Geometry Part A Final Exam
How are dilations and functions related?
wHEN DILATING A FIGURE, THE RULE (X, Y) -> (KX, KY), WHERE K IS THE SCALE FACTOR, CAN be seen as a function machine. Remember that (x, y) is the input and (kx, ky) is the output.
Prove the Triangle Proportionality Theorem
Δ ABC is intersected by line DE given DE || BC given ∡ABC ≅ ∡ADE Corresponding Angles Postulate ∡ACB ≅ ∡AED Corresponding Angles Postulate ΔABC ~ ΔADE Angle-Angle (AA) Similarity Postulate AD over AB equals AE over AC Definition of Similar Triangle
How do you construct parallel lines with a compass and straightedge?
• You are given a line. • Draw another line that intersects the first line. • Place a point on this second line. • Place the compass on the intersection point of the first line and the second line and swing an arc that crosses both lines. • Keeping the compass at the same width, place the compass on the point on the second line and swing an arc similar to the first—making sure you cross through the second line. • Go back to the given line and open the compass to the width of the intersection of the first arc and the two lines. • Keeping the compass at the same width, place the compass on the intersection point of the second line and the arc and swing another arc that intersects the first. • Mark the intersection of the two arcs. • Draw a line through the point on the second line and the intersection of the arcs, creating a third line which is parallel to the first.
Are reflections functions?
A reflection can also be seen as a function which is a type of input/output machine.
What is a reflection?
A reflection is a type of transformation that creates a mirror image of the pre-image across a line of reflection.
What is a rotation?
A rotation is a type of transformation in which the pre-image is turned around a fixed point, the center of rotation, in order to create the image. When an image is rotated around the origin of the coordinate plane, there are three rules that may be applied to the pre-image in order to find the postition of the image. Rotation of 90 degrees clockwise: (x, y) -> (y, -x) Rotation of 90 degrees counterclockwise: (x, y) -> (-y, x) Rotation of 180 degrees: (x, y) -> (-x, -y)
What is a Transformation?
A transformation describes the change in a figure. A figure could change size, orientation, location, or a combination of these.
What is the Alternate Interior Angles Theorem
If a transversal intersects two parallel lines, then alternate interior angles are congruent. When a transversal, the line that cuts through, intersects with two parallel lines, it creates eight angles, four of which are on the inside, or interior, of the parallel lines. The angles that are diagonal from each other are congruent, or have equal measures.
Converse of the Pythagorean Theorem
If the sum of the squares of the shorter sides is equal to the sqaure of the longest side, then the triangle has a right angle. Or, given c^2 = a^2 + b^2, where a, b, and c are the sides of a triangle, the triangle is a right triangle.
What is the Side-Angle-Side Similarity Postulate
If two or more triangles have corresponding congruent angles and the sides that make up these angles are proportional, then the triangles are similar.
How do you construct Equilateral Triangles, Squares, and Regular Hexagons inscribed in Circles?
*Look at "Geometry Unit 1 Lesson 4 Notes" saved in downloads
What are the properties of rectangles?
1. Both pairs of opposite sides are congruent and parallel. 2. The diagonals bisect each other. 3. Both pairs of opposite angles are congruent 4. Consecutive angles are supplementary 5. Contains four right angles 6. The diagonals are congruent
Conditional Statement
A conditional statement is an "if, then" statement that states if one event occurs, then the other will occur.
What is a line of symmetry?
A line of symmetry is a line drawn through a figure that creates two congruent halves.
What is the difference between a postulate and a theorem?
A postulate is considered a known fact. Undefined terms, defined terms, and postulates will help you prove theorem in geometry
What is a proof?
A proof is a series of steps or statements, with justifications, that verifies a theorem or relationship between two objects exists. You must be able to support each statement with a reason or justification as to why the statement is true. When designing a proof, it is very important not to skip steps, no matter how "easy" or "obvious" they may seem.
What is a translation?
A translation is a type of transformation that simply moves a figure from one location to another. The translation rule will tell you how the image has moved.
What is a vector?
A vector represents not only the magnitude of an object, but also its direction. In other words, vectors are used to describe objects in motion. A vector has a fixed lengt.
Prove the Converse of the Triangle Proportionality Theorem
AD over AB equals AE over AC ∡A ≅ ∡A ΔDAE ~ ΔBAC ∡ADE ≅ ∡ABC DE || BC Given Reflexive Property Side-Angle-Side (SAS) Similarity Postulate Corresponding Angles of Similar Triangles are Congruent Converse of Corresponding Angles Postulate
How do you prove the Alternate Interior Angles Theorem?
According to the given information in the image, segment AB is parallel to segment CD while angles AGF and EGB are vertical angles. Angle AGF is congruent to angle EGB by the Vertical Angles Theorem. Because angles EGB and EHD are corresponding angles, they are congruent according to the Corresponding Angles Theorem or Postulate. Finally, angle AGF is congruent to angle EHD by the Transitive Property of Equality.
Rectangles Are Parallelograms with Congruent Diagonals
According to the given information, quadrilateral RECT is a rectangle. By the definition of a rectangle, all four angles measure 90 degrees. Any two adjacent angles in rectangle RECT add up to180° since 90° + 90° = 180°. These adjacent angles are same-side interior angles because they lie inside, and on the same side of two lines intersected by a transversal. Therefore, segment ER is parallel to segment CT and segment EC is parallel to segment RT. Segment E R is parallel to segment C T by the Converse of the Same-Side Interior Angles Theorem.(This theorem states that when two angles are on the same side of two lines intersected by a transversal and the total of these angles is 180°, then the lines are parallel.) Quadrilateral RECT is then a paralellogram by definition of a paralellogram. Now, construct diagonals ET and CR. Since angleCTR and angleTRE both measure 90°, these angles are congruent according to the definition of congruence. Because RECT is a parallelogram, opposite sides are parallel. Therefore, one can say that segment ER is congruent to segment CT. Segment TR is congruent to itself by the Reflexive Property of Equality. There are now two corresponding, congruent sides (ER and CT with TR and TR) joined by a corresponding pair of congruent angles (angleERT and angleCTR). So the Side-Angle-Side (SAS) Theorem says triangleERT is congruent to triangleCTR. And because corresponding parts of congruent triangles are congruent (CPCTC), diagonals ET and CR are congruent.
How do you know that each of these constructions is valid when made with a compass and straightedge?
After you find the solution, ask yourself these questions to help provide a good explanation. How did you get that answer? What steps did you use to get there? Why did you choose to solve the problem the way you did? What other examples are similar to the problem you solved that helped you solve this problem? What were you thinking as you moved through each step of the problem? If someone questions you, what would you tell them to prove you solved the problem correctly?
What are orthocenters and altitudes?
Altitudes are drawn from each vertex of the triangle and intersect the opposite side , which may be extended, at a 90 degree angle. The orthocenter represents th point of intersection between the three altitudes. It can appear inside, outside, or on the triangle.
What are Angle Bisectors and Incenters?
Angle Bisectors are drawn from each vertex, splitting the angle at that vertex in two, to the opposite side. The incenter represents the point of intersection between the three angle bisectors. The incenter will always lie inside the triangle.
What is ASA?
Angle-Side-Angle: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.
How do you use methods of an indirect proof to determine whether a short proof is logically valid?
Assume the opposite of the conclusion (or prove statement) Reason logically to show the assumption leads to a contradiction of a known fact Conclude the assumption is false, which in turn proves the conclusion is true.
How do you construct segment bisectors?
Bisecting a segment You are given a segment with two endpoints. Place the compass on one of the endpoints and open the compass to a distance more than halfway across the segment. Swing an arc on either side of the segment. Keeping the compass at the same width, place the compass on the other endpoint and swing arcs on either side so that they intersect the first two arcs created. Mark the intersection points of the arcs and draw a line through those two points. The point where this new line crosses the given segment is the midpoint and divides the segment in half. When you bisect a segment, you also construct a perpendicular bisector. Justifying
Converse of Triangle Proportionality Theorem
If a line divides any two sides of a triangle proportionally, then the line must be parallel to the third side.
Triangle Proportionality Theorem
If a line is parallel to one side of a triangle and also intersects the other two sides, the line divides the sides proportionally.
What is the Alternate Exterior Angles Theorem?
If a transversal intersects two parallel lines, then alternate exterior angles are congruent. Alternate means opposite sides, so the angles are on opposite sides of the transversal. Exterior means outside, so more specifically, this theorem is stating that the opposite, outside angles are congruent, or have the same measure.
What is the Corresponding Angles Postulate?
If a transversal intersects two parallel lines, then corresponding angles are congruent. The corresponding angles are the angles on the same side of the transversal and in the same position. They also happen to have equal measures.
What is the Same-Side Interior Angles Theorem
If a transversal intersects two parallel lines, then same-side interior angles are supplementary.
Pythagorean Theorem
If a triangle has a right angle, then the sum of the squares of the shorter sides is equal to the square of the longest side. Or, given right triangle ABC, c^2 = a^2 + b^2, where c is the hypotenuse while a and b are the two legs.
What is the pieces of right triangle similarity theorem?
If an altitude is drawn from the right angle of a right triangle, the two smaller triangle created are similar to one another and to the larger triangle. An altitude in a right triangle creates two smaller right triangles. All three of the triangles, the big one and the two smaller ones inside the big one, are similar to one another.
Side-Side-Side Similarity Postulate
If the sides of one triangle are proportional to the sides of a second triangle, then the triangles are similar.
What is the Hypotenuse-Leg Theorem?
If two right triangles have congruent hypotenuses and corresponding, congruent legs, the two right triangles are congruent.
What is the Isosceles Triangle Theorem and how do you prove it?
If two sides of a triangle are congruent, then the angles opposite those sides are congruent. The two base angles are equal in measure. But remember, the base angles do not have to be at thebottom of the triangle. They are the angles adjacent to the unequal side of the triangle. In an isosceles triangle where all three sides are equal, also known as an equilateral triangle, then all three angles will be equal as well. Line RS is congruent to line ST according to the given information. Using a compass and straightedge, construct line US as an angle bisector of angle RST. Angle RSU is congruent to angle UST by the definition of an angle bisector. Line US is congruent to line US by the Reflexive Proporty of Equality. Triangle RSU is congruent to triangle TSU by the Side-Angle-Side Postulate. Therefore, angle TRS is congruent to angle STR by CPTCTC (Corresponding Parts of Congruent Triangles are Congruent.)
What is the Midsegment of a Triangle Theorem and how do you prove it?
It states that a segment connecting the midpoints of two sides of a triangle is parallel to the third side and its length is equal to half the length of the third side. Two-column Proof Statement Reason Point J is the midpoint of RU. Point O is the midpoint of UN. Given Graph ΔRUN on the coordinate plane. Let R be (0, 0), U be (x1, y1), and N be (x2, 0). by Labeling x sub 1 plus x sub 2 all over 2, y sub 1 plus y sub 2 all over 2 Midpoint Formula J:(x1 plus zero over two comma y1 plus zero over two) Substitution O:(x1 plus x2 over two comma y1 plus zero over two) Substitution y2 minus y1 over x2 minus x1 Slope Formula mJO = [ (y1 plus zero over 2) minus (y1 plus zero over two) ] over [ (x1 plus x2 over two) minus (x1 plus zero over two) ] Substitution mJO = [zero over two] divided by x2 over two Subtraction mJO = [zero over two] divided by x2 over two = 0 Division mRN = zero minus zero over x2 minus zero Substitution mRN = zero over x subscript two Subtraction mRN = 0 Division mJO = mRN Transitive Property of Equality JOparallelRN Definition of Parallel Lines
What is the Converse of the Isosceles Triangle Theorem and how do you prove it?
It states that given two congruent angles of a triangle, the sides opposite these angles will be congruent. In ΔOLN, ∠OLN is congruent to ∠LNO by the given information. Construct OE as a perpendicular bisector to LN. The measures of ∠LEO and ∠NEO are both 90° by construction of a perpendicular bisector. By the transitive property of equality ∠LEO is congruent to ∠NEO. Similarly, LE is congruent to EN by the definition of a perpendicular bisector. Consequently, ΔOLE is congruent to ΔONE by the Angle-Side-Angle (ASA) Postulate. Because corresponding parts of congruent triangles are congruent (CPCTC), OL is congruent to ON.
What are Perpendicular Bisectors and Circumcenters?
Perpendicular Bisectors intersect each side of the triangle at a 90 degree angle at that side's midpoint. The circumcenter represents the point of intersection between the three perpendicular bisectos. It can appear inside, outside, or on the triangle.
Paragraph Proof
Proofs in which you write the steps and justifications using complete sentences and in paragraph form.
Prove triangle AED is similar to triangle BEC
Prove: ΔAED ~ ΔBEC Statements Reasons DC intersects AB at point E given AD || CB given ∡AED ≅ ∡BEC Vertical Angles Theorem ∡DAE ≅ ∡CBE Alternate Interior Angles Theorem ΔAED ~ ΔBEC Angle-Angle Similarity Postulate
Prove that diagonals of a parallelogram bisect each other
Quadrilateral PARL is a parallelogram (given). Line PA is parallel to line LR and line AR is parallel to line PL (definition of a parallelogram). Angle APR is congruent to angle LRO (Alternat Interior Angles Theorem) Angle PAL is congruent to angle RLA (Alternate Interior Angles Theorem). Line PA is congruent to line RL (Proprty of a Parallelogram (Opposite Sides are Congruent)). Triangle APO is congruent to triangle LRO (ASA theorem). Line PO is congruent to line RO and line AO is congruent to line LO (CPCTC) Diagonals PR and AL bisect one another (definition of a bisector)
Prove that Opposite Sides of Parallelograms are Congruent
Quadrilateral PARL is a parallelogram (given9). Draw line PR (by contruction) Line PA is parallel to line LR and line AR is parallel to line PL (Definition of a Parallelogram). Angle APR is congruent to angle PRL (Alternate Interior Angles Theorem). Line PR is congruent to line PR (Reflexive Property of Equality) Angle PRA is congruent to angle RPL (Alternate Interior Angles Theorem). Triangle APR is congrueent to triangle LRP (Angle-Side-Angle Theorem). Line AP is congruent to line LR and line AR is congruent to line LP (CPCTC)
What is SAS?
Side-Angle-Side: if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
What is SSS?
Side-Side-Side: If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.
How do you determine if polygons are similar?
Similar polygons are polygons that have congruent angles and corresponding sides that are proportional to one another
Is a translation a function?
Yes, a translation can also be seen as a function which is a type of input/output machine.
How do you construct perpendicular lines with a point on a given line using a compass and straightedge?
You are given a line and a point on the line. Mark the intersection point of these two lines. Place the compass on this intersection and swing an arc on either side of this point, making sure to cross the given line on both sides. Mark the two intersection points of the given line and two arcs. Place the compass on one of these intersection points and draw two arcs, one on either side of the given line. Keeping the compass at the same width, place the compass on the second intersection point and draw two arcs, one on either side of the line. Make sure that the two arcs on either side of the line intersect one another. Mark the intersection point of the two arcs on either side of the line. Draw a line through these two intersection points. This new line is perpendicular to the given line.
How do you construct perpendicular lines using a point not on a line?
You are given a line and a point that is not on the line. Place the compass on the given point and open it so that the width of the compass is more than the distance from the given point to the given line. Swing two arcs that intersect the given line. Mark the two intersection points of the given line and two arcs. Place the compass on one of these intersection points and draw two arcs, one on either side of the given line. Keeping the compass at the same width, place the compass on the second intersection point and draw two arcs, one on either side of the line. Make sure that the two arcs on either side of the line intersect one another. Mark the intersection point of the two arcs on either side of the line. Draw a line through these two intersection points. This new line is perpendicular to the given line.
How do you construct congruent segments?
You are given a segment with two endpoints. Draw a ray with one endpoint. Open the compass to the width of the given segment. Place the compass on the new ray's endpoint and swing an arc that intersects the ray. The intersection point of the ray and arc is the second endpoint that makes the new line segment congruent to the given one.
What is the angle of rotation?
The angle of rotation is the number of degrees the figure must rotate before it looks like itself again. The angle of rotation can be found by dividing 360 degrees by the order of rotation
What is a centroid?
The centroid represents the point of intersection between the three medians. It will always lie inside the triangle
What is invese
The inverse of a statement is the statement in the same order with both parts negated
How do you construct angles?
You are given an angle. Draw a ray with one endpoint. This endpoint will be the vertex of the new angle. Place the compass on the vertex of the given angle and swing an arc that intersects both rays of the given angle. Place the compass on the vertex of the new angle and swing an arc similar to the first one you created. Open the compass to the width of the intersection points of the rays and arc of the given angle. Place the compass on the intersection point of the ray and arc of the new angle and swing another arc that intersects the first. Draw a ray through the new vertex and the intersection point of the two arcs. This second ray creates an angle that is congruent to the given one.
Proving the Midsegment of a Triangle Theorem, Part2
Two-column Proof Statement Reason JO parallel RN R: (0, 0) U: (x1, y1) N: (x2, 0) J is the midpoint of RU O is the midpoint of UN J:(x1 plus zero over two comma y1 plus zero over two) O:(x1 plus x2 over two comma y1 plus zero over two) Given (from earlier proof) d = square root of (x2 minus x1) squared plus (y2 minus y1) squared Distance Formula dJO = square root of (x1 plus x2 over two minus x1 plus zero over two) squared plus (y1 plus zero over two minus y1 plus zero over two) squared Substitution dJO = square root of (x2 over two) squared plus (zero ovdr two) squared Subtraction dJO = square root of (x2 over two) squared plus zero squared Division dJO = square root of (x2 over two) squared Additive Identity dJO = x2 over two = one over twox2 Square Root Property of Equality dRN = square root of (x2 minus zero) squared plus (zero minus zero) squared Substitution dRN = square root of x2 squared plus zero squared Subtraction dRN = square root of x2 squared Additive Identity dRN = x2 Square Root Property of Equality dJO = one over twox2; dRN = x2 dJO = one over two(dRN) Substitution
How do you determine if two triangles are similar?
You can identify similar triangles using the Angle-Angle Similarity Postulate: If two triangles have congruent angles, the triangle are similar. Similar triangles have corresponding parts that form a proportion with their corresponding sides.
How many specific parts are required to declare two triangles congruent using SSS, SAS, or ASA?
You need a minimum of three specific parts to declare two triangles congruent using those theorems.
What are the three types of proofs?
Two-column: statements are listed in the left column. reasons or justifications are listed in the right column Paragraph: To help justify each statement, use terms such as: because, since, by Flow Chart: statements are written in boxes. reasons or justifications are underlined underneath the statement they support. Use arrows to connect appropriate statements together.
How do defined terms and undefined terms relate to each other?
Undefined terms will be used as a foundational element in defining other "defined" terms. The undefined terms include point, line, and plane. The defined terms discussed so far include angle, circle, perpendicular line, parallel line, and line segment.
How do you dilate a figure on the coordinate plane?
When a figure is dilated from the origin, each ordered pair of the image may be found according to the rule (x, y) -> (kx, ky) where k is the scale factor.
What is Congruency?
When two figures are congruent, that means they are the exact same shape and their measures are equal. To symbolically represent congruency use thecongruent signsign.
How are images reflected across the y-axis
(x, y) -> (-x, y)
Biconditional statement
An "if and only if" statement that states that both events are dependent on each other occurring
How are images reflected across the x-axis?
(x, y) -> (x, -y)
How are images reflected across the line y = x?
(x, y) -> (y, x)
How do you construct angle bisectors using tools such as a compass and straightedge?
Bisecting an angle You are given an angle. Place compass on the vertex of the angle. Swing an arc that intersects both rays of the angle. Mark the intersection points of the rays and arc. Place the compass on one of those intersection points and draw an arc inside the angle. Keeping the compass at the same width, place the compass on the second intersection point and swing an arc that intersects the first. Mark the intersection point of the two arcs and draw a ray from the vertex through this intersection point.
What is the Concurrency of the Medians of a Triangle
Construct the midpoint of WY and label it point A. By definition of a midpoint, WA is congruent to AY. Construct the midpoint of XY and label it point B. By definition of a midpoint, XB is congruent to BY. Construct XA. Construct WB. XA and WB must intersect in exactly one point by the Intersecting Lines Postulate. Let's call this point D. Construct YD. YD must intersect WX in exactly one point by the Intersecting Lines Postulate. Construct point P on YD so that YD is congruent to DP. Construct WP and PX. AD || WP by the Midsegment Theorem.DX is an extension of AD because they lie on the same line. Therefore, DX || WP by Substitution. DB || PX by the Midsegment Theorem. WD is an extension of DB because they lie on the same line. Therefore, WD || PX. WDXP is a parallelogram because opposite sides are parallel, which is a property of parallelograms. Because the diagonals of a parallelogram bisect one another, DC is congruent to CP and WC is congruent to CX. Point C represents the midpoint of WX by definition of a midpoint. By definition of a median, this makes YC a third median of △WXY that intersects the other two medians at point D. Consequently all three medians of △WXY are concurrent.
What is converse
Converse of a statement is the statement in the opposite order
What relationships are formed by corresponding angles of similar polygons?
Corresponding angles of similar polygons will always have congruent measures.
What are corresponding parts?
Corresponding parts of a triangle can be labeled for all six parts of two triangles
How are images reflected across a horizontal or vertical line?
Count the distance between each point on the figure and the line of reflection. Then, for each point, count this same distance from the line of reflection to find the corresponding point.
Proving the Pythagorean Theorem with Similar Triangles
Draw right triangle ABC (by construction). Draw altitude line CD with length h (by construction). Let line AC = b, line CB = a, line AB = c, line AD = x, and line DB = y (by labeling). c/a = a/y and c/b = b/x (Pieces of Right Triangles Similarity Theorem). B squared = cx (cross product property). a squared = cy (cross product property). a^2 + b^2 = cy +b^2 (Addition property of equality). a^2 + b^2 = cy + cx (substitution). a^2 + b^2 = c(y+x) (Distributive property of equality). a^2 + b^2 = c(c). y + x = c (segment addition postulate). a^2 + b^2 = c(c) (substitution). a^2 + b^2 = c^2
What relationships are formed by corresponding sides of similar polygons?
Each pair of corresponding sides of similar polygons will have equal ratios.
How do you prove that vertical angles are congruent?
Slide 1 You are given an image of two intersecting segments and asked to prove that angle 1 is congruent to angle 3. By looking at the image, you can see that the two intersecting segments form four angles. The first step is to point out that two angles that form a straight line sum to 180 degrees by the definition of supplementary angles. Therefore, angle 1 plus angle 2 is equal to 180. Slide 2 The second step is very similar to the first. Another pair of two angles that form a straight line are angles 2 and 3. And, by definition of supplementary angles, we know that angle 2 plus angle 3 is equal to 180. Slide 3 By the transitive property of equality, since both the sum of angles 1 and 2 and the sum of angles 2 and 3 equal 180 degrees, you know that these sums must be equal to one another. So the sum of angle 1 and angle 2 is equal to the sum of angle 2 and angle 3. Slide 4 By the subtraction property of equality, angle 2 can be subtracted from both sides of the equation, proving that the measures of angle 1 and 3 are equal. Therefore, angle 1 is congruent to angle 3.
Prove that Bisecting Diagonals in a Quadrilateral can show the Quadrilateral is a parallelogram
Statement Justification In Quadrilateral, PARLL, diagonals PR and AL bisect one another. (given). Line PO is congruent to line RO and line AO is congruent to line LO (definition of a bisector). Angle POA is congruent to angle ROL and angle ROA is congruent to angle POL by the vertical Angles Theorem. Triangle POA is congruent to triangle ROL and triangle ROA is congruent to triangle POL (SAS Theorem)/ Angle APO is congruent to angle LRO, angle OAP is congruent to angle OLR, and angle OAR is congruent to angle OLD, and angle ARO is congruent to angle LPO (CPCTC). Line PA is parallel to line LR and line AR is parallel to line PL. (Converse of the Alternate Interior Angles Theorem). Quadrilateral PARL is a parallelogram (Definition of a parallelogram).
How do you prove the Corresponding Angles Postulate?
Statements Reasons segment A B is parallel to segment C D Given Points E, G, H, and F all lie on the same line. Given mangleEGF = 180° Definition of a Straight Angle mangleAGE + mangleAGF = mangleEGF Angle Addition Postulate mangleAGE + mangleAGF = 180° Substitution Property of Equality mangleCHE + mangleAGF = 180° Same-Side Interior Angles Theorem mangleAGE + mangleAGF = mangleCHE + mangleAGF Substitution Property of Equality mangleAGE + mangleAGF - mangleAGF = mangleCHE + mangleAGF - mangleAGF mangleAGE = mangleCHE Subtraction Property of Equality angleAGE ≅ angleCHE Definition of Congruency
Prove that opposite angles of parallelograms are congruent
Statements Justifications Quadrilateral PARL is a parallelogram Given segment P A is parallel to segment L R and segment A R is parallel to segment P L Definition of a Parallelogram Extend each side of the parallelogram and place a point on each extension by Construction anglePAR ≅ angleEPA Alternate Interior Angles Theorem angleEPA ≅ angleRLP Corresponding Angles Theorem anglePAR ≅ angleRLP Transitive Property of Equality angleAPL ≅ anglePLW Alternate Interior Angles Theorem anglePLW ≅ angleLRA Corresponding Angles Theorem angleAPL ≅ angleLRA Transitive Property of Equality
How do you prove that a point on a perpendicular bisector is equidistant from the endpoints of the segment it intersects?
Statements Reasons line CD is a perpendicular bisector of segment AB Given segment AE is congruent to segment EB Definition of a Perpendicular Bisector angleAEC = 90° angleBEC = 90° Definition of a Perpendicular Bisector angleAEC ≅ angleBEC Definition of Congruence segment CE is congruent to segment CE Reflexive Property of Equality Draw segment AC and segment CB by Construction using a Straightedge ΔAEC ≅ ΔBEC Side-Angle-Side (SAS) Postulate segment A is congruent to segment CB Corresponding Parts of Congruent Triangles are Congruent (CPCTC) CA = CB Definition of Congruence Point C is equidistant from points A and B. Definition of Equidistant
What are the types of angles?
Straight angles are lines or line segments, and the measurement is 180 degrees Acute angles are angles that measure less than 90 degrees Obtuse angles are angles that measure more than 90 degrees, but less than 180 degrees Right angles are angles that measure exactly 90 degrees
Two-Column Proofs
Tables with two columns, the first is the mathematical statement, the second is the justification
What is contrapositive
The Contrapositive of a statement is the statement in the opposite order with both parts negated
Prove the Converse of the Pythagorean Theorem
The length of BC is a, the length of AC is b, and the length of AB is c, according to the given information. Construct right triangle SIT where legs ST and TI are congruent to legs AC and CB of triangle ABC, respectively.In other words, TI is congruent to CB and ST is congruent to AC by construction. ST has length i, TI has length s, and IS has length t. Because ΔSIT was constructed as a right triangle, t2 = s2 + i2 by the Pythagorean Theorem. Substitute a for s and b for i in the Pythagorean Theorem so t2 = a2 + b2. According to the given information, c2 = a2 + b2. By the Substitution Property of Equality, t2 = c2. Take the square root of both sides of the equation, according to the Square Root Property of Equality, to reach the equation t = c. By the , ΔABC is congruent to ΔSIT. In conclusion, ∠ACB is a right angle by CPCTC.
What is a transversal?
The line that intersects parallel lines, creating multiple pairs of angles with a similar relationship to one another.
What is negation
The negation of a statement is the opposite of that statement
What is the order of rotation?
The number of times the image looks like the pre-image is known as the order of rotation.
What is the name of the original figure in a transformation and the new figure?
The original figure's name is pre-image, the new figure's name is image. An arrow is used to indicate how a characteristic of the figure has changed. For example, A→A' means A maps to A'. These are corresponding points.
What is a scale factor?
The scale factor is the constant by which a figure (or the dimensions of a figure) increases or decreases. If the scale factor is a number greater than one, the figure will be enlarged. If the scale factor is a number between zero and one, the figure will be reduced in size.
What is the Triangle Sum Theorem and how do you prove it?
The sum of the measures of the angles in a triangle is 180 degrees. Two-column Proof Statement Reason Draw AN parallel to TR by Construction m∠AIT + m∠RIT + m∠RIN = m∠AIN Angle Addition Postulate ∠AIT ≅ ∠ITR Alternate Interior Angles ∠RIN ≅ ∠TRI Alternate Interior Angles m∠ITR + m∠RIT + m∠TRI = m∠AIN Substitution ∠AIN = 180° Definition of a Straight Angle m∠ITR + m∠RIT + m∠TRI = 180° Substitution
What is a vertex angle?
The third angle of a triangle that is not congruent to the other two is called the vertex angle
What is the Median?
They are drawn from the midpoint of each side of the opposite vertex
What is Rotational symmetry?
This happens when a figure turns around its center point and the image looks like the pre-image before turning a full 360 degrees.
