Chapter 4
"If a point lies on the Perpendicular Bisector of a line segment, then it is equidistant to the endpoints of the segment."
Complete the following sentence. "If a point lies on the ____________ of a line segment, then it is equidistant to the endpoints of the segment."
"If a point lies on the bisector of an angle, then it is equidistant to the sides of the angle."
Complete the following sentence. "If a point lies on the ____________ of an angle, then it is equidistant to the ________ of the ________."
"The circumcenter of a triangle is equidistant to the vertices of a triangle. "
Complete the following sentence. "The circumcenter of a triangle is equidistant to__________________ of a triangle. "
"The incenter of a triangle is equidistant to the sides of a triangle."
Complete the following sentence. "The incenter of a triangle is equidistant to the ______ of a triangle."
You will construct the circumcenter. You will need to construct the perpendicular bisectors of AB, BC, and/or AC. Then find the intersection of the perpendicular bisectors to find the point that is equidistant to A, B, and C.
Given 3 points A, B, and C, construct the point which is equidistant from A, B, and C. What will you construct?
You will construct the incenter. You will need to construct the angle bisectors of <A, <B, and/or <C Then find the intersection of the angle bisectors to find the point that is equidistant to AB, BC, and AC.
Given 3 points A, B, and C, construct the point which is equidistant from AB, BC, and AC. What will you construct?
3
How many altitudes does a triangle have?
3
How many medians does a triangle have?
In an isosceles triangle, the NON-congruent side is called the base.
In an isosceles triangle, the NON-congruent side is called the ________.
In an isosceles triangle, the angle that is not a base angle is called the vertex or apex angle.
In an isosceles triangle, the angle that is not a base angle is called the ________ angle.
In an isosceles triangle, the congruent sides are called the legs.
In an isosceles triangle, the congruent sides are called the ________.
1. If 2 sides of a triangle are congruent, then the angles opposite the sides are congruent. (Base angles are congruent) 2. If 2 angles of a triangle are congruent, then the sides opposite the angles are congruent.
Name the 2 theorems relating to isosceles triangles
Line segment AH, AB, and BC
Name the altitudes shown in the diagram
Line segment MD, BC, and AC
Name the altitudes shown in the diagram
line segment AB
Name the base of triangle ABC
Hypotenuse.
Provide another name for line segment AB
SSS postulate; triangle EGF = triangle MLK Angles Sides 1) <E =<M; 4) EG = ML 2) <G =<L 5) GF = LK 3) <F = <K 6) EF = MK
State the postulate that can be used to prove the triangles congruent. Then, use the diagram to write the congruence and name the congruent corresponding parts.
The diagram shows SSA or ASS. The angle and the two sides shown in red indicate that these pieces of the triangle are "set." Thus, the set pieces can form one triangle triangle CBH, and then form a second triangle, triangle CBG, by moving the segment of BH along CH. This shows that if two sides and a non-included angle of one triangle, are congruent to the corresponding two sides and a non-included angle of another triangle, then the triangles are NOT congruent.
The following diagram disproves which theorem for triangle congruence? Explain
Angles Sides 1) <F =<R; 4) EF= RS 2) <D =<D 5) ED=DS 3) <E = <S 6) FD = DR
Use the congruence shown in the diagram to name the congruent corresponding parts.
Angles Sides 1) <Y =<C; 4) YZ = ZC 2) <Z =<Z 5) XZ = XZ 3) <X = <X 6) YX = XC
Use the congruence shown in the diagram to name the congruent corresponding parts.
Not congruent
Use the diagram to determine whether the triangles are congruent. If yes, name the postulate that justifies their congruence. Then, complete the congruence. If no, then state, "not congruent."
Yes Triangle STR = triangle QTP By SAS
Use the given information to state whether the triangles are congruent. If yes, name the triangle congruence and then identify the Theorem or Postulate that would be used to prove the triangles congruent. If the triangles cannot be proven congruent, state "not possible."
Yes. Triangle EGH = FGH By SAS (directly) By ASA (indirectly, using the fact that triangle EGF is isosceles Thus, in an isosceles triangle base angles are congruent, which means <E = <F) By AAS (even more indirectly, using the fact that <E = <F and <EGH = <FGH. Therefore, if 2 angles of a triangle are congruent to 2 angles of another triangle, then the 3rd angles are congruent. This now means we also have AAS, two angles a and a non-included side congruent to two angles and a non-included side of another triangle.)
Use the given information to state whether the triangles are congruent. If yes, name the triangle congruence and then identify the Theorem or Postulate that would be used to prove the triangles congruent. If the triangles cannot be proven congruent, state "not possible."
Figures with the same size and the same shape
What are congruent figures?
They are the parts that are congruent to one another in each congruent figure. Corresponding parts share the same position relative to the figure.
What are corresponding parts of congruent figures?
1. Side-Side-Side Postulate (SSS) 2. Side-Angle-Side Postulate (SAS) 3. Angle-Side-Angle Postulate (ASA) 4. Angle-Angle-Side Theorem (AAS or SAA) 5. Hypotenuse-Leg Theorem (HL)
What are the 5 methods that can be used for proving triangles congruent?
Corresponding Parts of Congruent Triangles are Congruent
What does CPCTC stand for?
Medians: line segment IT, line segment TO Altitudes: line segment IU, line segment JP Perpendicular Bisectors: line segment SR
Based on the markings in the figure, name all of the altitudes, medians, and perpendicular bisectors shown in Δ IJK.
Median: line segment PS Altitude: line segment RT Angle Bisector: Ray QU
Based on the markings in the figure, name an altitude, a median, and an angle bisector of Δ QRS.
1. Construct the angle bisectors of at least 2 of the angles of the triangle. 2. Extend the angle bisectors so they intersect. The intersection point is called the incenter. 3. Construct a line through the incenter that is perpendicular to any of the 3 sides of the triangle. 4. The distance from the incenter to where the perpendicular line intersects the side of the triangle is the size of your circle (the radius of your circle) 4. Complete the circle. The circle should barely intersect 3 points on the triangle.
Explain the process required to construct an incircle
1. Construct the perpendicular bisectors of at least 2 of the segments of the triangle. 2. Extend the perpendicular bisectors so they intersect. The intersection point is called the circumcenter. 3. Use the circumcenter as the center of your circle and extend your compass to touch ANY of the vertices of the triangle. This marks the size (radius) of your circle. 4. Complete the circle. The circle should pass through all three vertices.
Explain the process required to construct the circumcircle of a triangle.
their corresponding parts are congruent.
Finish the following sentence.... "if two figures are congruent, then...."
We are implying that the shortest distance is taken from point L to the sides of triangle JIK, which forms a 90 degree angle.
In saying that point L is equidistant to the sides of triangle JIK, what is implied?
Angle-Angle-Angle (AAA) AAA doesn't work because you could have two triangles with the exact same angle measures, but with different side lengths. For example, you could have two equiangular triangles with different side lengths. Side- Side-Angle (SSA or ASS) Doesn't work because you could have a set side, next to a set side and then a set angle. The third side, if long enough, could intersect the first side in two different locations.
Name the two theorems that are not a guarantee for proving triangles congruent. Explain why they do not work.
Yes, SAS, Triangle ABE = triangle CDE
Use the diagram to determine whether the triangles are congruent. If yes, name the postulate that justifies their congruence. Then, complete the congruence. If no, then state, "not congruent."
Yes, SAS, Triangle ACD = triangle BCD
Use the diagram to determine whether the triangles are congruent. If yes, name the postulate that justifies their congruence. Then, complete the congruence. If no, then state, "not congruent."
Yes, SSS, Triangle OEG.
Use the diagram to determine whether the triangles are congruent. If yes, name the postulate that justifies their congruence. Then, complete the congruence. If no, then state, "not congruent."
Angles Sides 1) <E =<J; 4) EF = FD 2) <F=<I 5) JI = IK 3) <D = <K 6) ED = JK
Use the diagram to name the congruent corresponding parts.
Angles Sides 1) <M =<A; 4) AB = ML 2) <L =<B 5) BC = LN 3) <N = <C 6) AC = MN
Use the diagram to name the congruent corresponding parts.
SSS postulate; triangle CBD = triangle CLD Angles Sides 1) <C =<C; 4) CB = CL 2) <B =<L 5) CD = CD 3) <D = <D 6) BD= DL
Use the diagram to: a) state the postulate that can be used to prove the triangles congruent b) write the congruence c) name the congruent corresponding parts.
No Not possible, SSA or ASS only works with right triangles
Use the given information to state whether the triangles are congruent. If yes, name the triangle congruence and then identify the Theorem or Postulate that would be used to prove the triangles congruent. If the triangles cannot be proven congruent, state "not possible."
Yes Triangle CAB = triangle DBA By SSS t
Use the given information to state whether the triangles are congruent. If yes, name the triangle congruence and then identify the Theorem or Postulate that would be used to prove the triangles congruent. If the triangles cannot be proven congruent, state "not possible."
Angle-Side-Angle Postulate state?
What does ASA stand for?
If two angles and a NON-INCLUDED side of one triangle are congruent to the two corresponding angles and the corresponding NON-INCLUDED SIDE of another triangle, then the triangles are congruent.
What does Angle-Angle-Side Theorem Of Congruence state?
Corresponding parts of congruent triangles are congruent.
What does CPCTC stand for?
Side-Angle-Side Postulate state?
What does SAS stand for?
Side-Side-Side Postulate
What does SSS stand for?
If two angles and an included side of one triangle are congruent to two angles and an included side of another triangle, then the two triangles are congruent.
What does the Angle-Side-Angle Postulate state?
If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.
What does the Hypotenuse-Leg Theorem of Congruence state?
If two sides and an included angle of one triangle are congruent to two sides and an included angle of another triangle, then the two triangles are congruent.
What does the Side-Angle-Side Postulate state?
If three sides of a triangle are congruent to three sides of another triangle, then the two triangles are congruent.
What does the Side-Side-Side Postulate state?
A median of a triangle is a line segment joining a vertex to the midpoint of the opposing side.
What is a median?
A line, ray, or segment that divides a line segment into two equal parts. It also makes a right angle with the line segment.
What is a perpendicular bisector?
An altitude of a triangle is a line segment through a vertex and perpendicular to (i.e. forming a right angle with) a line containing the opposite side of the triangle.
What is an altitude of a triangle?
A triangle with at least 2 congruent sides
What is an isosceles triangle?
The point at which the perpendicular bisectors of the sides of a triangle intersect.
What is the circumcenter of a triangle?
The incenter of a triangle is the point where the three angle bisectors meet.
What is the incenter of a triangle?
