unit 4

n-gon

A polygon with n sides

legs of a trapezoid

A trapezoid has two bases, each of which is one of the parallel sides. The other two sides that aren't parallel to each other are called the trapezoid's legs.

area of a trapezoid

A trapezoid is a 4-sided figure with one pair of parallel sides. For example, in the diagram to the right, the bases are parallel. To find the area of a trapezoid, take the sum of its bases, multiply the sum by the height of the trapezoid, and then divide the result by 2, The formula for the area of a trapezoid is: or.

opposites side of quadrilteral

Another quadrilateral that you might see is called a rhombus. All four sides of a rhombus are congruent. Its properties include that each pair of opposite sides is parallel, also making it a parallelogram.

base of a triangle

Any side can be a base, but every base has only one height. The height is the line from the opposite vertex and perpendicular to the base. The illustration below shows how any leg of the triangle can be a base and the height always extends from the vertex of the opposite side and is perpendicular to the base.

area of a paraellelogram

Area is 2-dimensional like a carpet or an area rug. A parallelogram is a 4-sided shape formed by two pairs of parallel lines. Opposite sides are equal in length and opposite angles are equal in measure. To find the area of a parallelogram, multiply the base by the height.

perimeter of a polygon

Correct answer: Explanation: To find the perimeter of a regular hendecagon you must first know the number of sides in a hendecagon is 11. When you know the number of sides of a regular polygon to find the perimeter you must multiply the side length by the number of sides.

theorem 10.4

Dec 4, 2017 - Chapter 10 Class 9 Circles. ... Next: Theorem 10.5→. ... Given : A circle with center at O. AB is chord of circle & OX bisects AB i.e.

theorem 6.18

Definition. The mid segment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. Term. Kite Diagonal Theorem (Theorem 6.18) Definition.Jan 25, 2013

midsegment of a trapezoid

Explanation: A midsegment of a trapezoid is a segment that connects the midpoints of the two non-parallel sides of a trapezoid. This segment has two special properties. It is parallel to the bases of the trapezoid. The length of the midsegment is the average of the lengths of the two bases.

theorem 10.8

Feb 15, 2018 - Theorem 10.8 The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. Given : A circle with center at O. Arc PQ of this circle subtends angles POQ at centre O & ∠ PAQ at a point A remaining part of circle. To Prove : ∠POQ = 2∠PAQ ...

theorem 6.7

Geometry Theorem 6.7 Proof. DrawingPad. Theorem 6.7: If a ray bisects an angle of a triangle, then it divides the opposite sides into segments whose lengths are proportional to the lengths of the other two sides (cf. Elements VI.3). - dcp ...

theorem 6.2

If line a divides any two side of a triangle in the same ratio, then the line is parallel to third side. Given :- Δ ABC and a line DE intersecting AB at D and AC at E, such that AD DB = AE EC To Prove :- DE ∥ BC Construction

theorem 6.1

If two parallel lines are transected by a third, the alternate interior angles are the same size. ... Thus, the line determined by D' and B is a line through B

hexagon

In geometry, a hexagon is a six sided polygon or 6-gon. The total of the internal angles of any hexagon is 72

area of a square

In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length.

theorem 10.1

In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. ... If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. ... In the same circle, or in congruent circles ...

theorem 6.21

May 28, 2014 - Coxeter- Theorem 6.21. Coxeter, Projective Geometry (Second Edition), page 53. View Worksheet. Share; Download. What do you want to download? You can either download the .ggb ... Coxeter- Theorem 6.23. May 28, 2014 - 10:52 PM LexiPasi. Coxeter- Figure 6.3A. May 28, 2014 - 11:34 PM LexiPasi ...

theorem 10.3

Nov 23, 2017 - Chapter 10 Class 9 Circles. ... Given : C is a circle with center at O. AB is a chord such that OX ⊥ AB To Prove : OX bisect chord AB i.e. AX = BX Proof : In ∆OAX = ∆OBX ∠OXA = ∠OXB OA = OB OX = OX ∴ ∆OAX ≅ ∆OBX AX = BX Hence, Proved.

theorem 10.10

Nov 23, 2017 - Theorem 10.10 If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle (i.e. they are concyclic). Given : A, B, C and D are 4 points (no 3 are collinear) AB subtends equal angles at C and D i.e. ...

theorem 10.12

Nov 23, 2017 - Theorem 10.12 If the sum of a pair of opposite angles of a quadrilateral is 180°, the quadrilateral is cyclic. Given : ABCD is 𝑎 quadrilateral such that ∠BAC + ∠BDC = 180° Prove : ABCD is a cyclic quadrilateral Proof : Since A, B, C are non-collinear One circle passes through three collinear points Let us ...

theorem 10.2

Nov 23, 2017 - Theorem 10.2 If the angles subtended by the chords of a circle at the center are equal, then the chords are equal. Given : A circle with center O. AB and CD are chords that subtend equal angles at center i.e. ∠AOB = ∠DOC To Prove : AB = CD Proof : In ΔAOB & ΔDOC OA = OD ∠AOB = ∠DOC OB = OC ...

theorem 10.6

Nov 23, 2017 - Theorem 10.6 Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres). Given : A circle with center at O. AB and CD are two equal chords of circle i.e. AB = CD & OX and OY are perpendiculars to AB & CD respectively. To Prove : OX = OY Proof : Now, given that AB = CD ...

theorem 10.7

Nov 23, 2017 - Theorem 10.7 Chords equidistant from the centre of a circle are equal in length. Given : C is 𝑎 circle with center at 0. AB and CD are two Chords of the circle where OX is distance from center i.e. OX ⊥ AB & OY is distance from center i.e. OY ⊥ CD & OX = OY To Prove : AB = CD Proof : In ∆AOX and ∆CDY ...

theorem 10.9

Nov 23, 2017 - Theorem 10.9 Angles in the same segment of a circle are equal. Given : A circle with center at O. Points P & Q on this circle subtends angles ∠ PAQ and ∠ PBQ at points A and B respectively. And chord PQ subtends ∠ POQ at the center To Prove : ∠PAQ = 2∠PBQ Proof : From Theorem 10.8: Angle ...

theorem 6.3

Nov 27, 2017 - Theorem 6.3 (AAA Criteria) If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangle are similar. ... Proof :- In ∆ABC and ∆DPQ AB = DP AC = DQ ∠A = ∠D ⇒ ∆ABC ≅ ∆DPQ ⇒ ∠B ...

base of a trapezoid

Opposite sides of an isosceles trapezoid are the same length (congruent). The angles on either side of the bases are the same size/measure

theorem 6.4

Proof: Theorem 6.2 says that a), b), and c) are equivalent. Theorem 6.1 states that d) implies a). So all that is needed is to show that c) implies d). Suppose the two lines are not parallel. If they are not parallel, then they meet at a single point by Theorem 1.6. That point will be on one side of the transverse line or the other.

base angle of a trapezoid

Properties of the sides of an isosceles trapezoid: The bases (top and bottom) of an isosceles trapezoid are parallel. Opposite sides of an isosceles trapezoid are the same length (congruent). The angles on either side of the bases are the same size/measure (congruent).

theorem 6.22

Prove the following variation of Theorem 6.22: If f : V → V is a rigid motion on a finite-dimensional real inner product space V, then there exists a unique orthogonal operator T on V and a unique translation g on V such that f = T ? g. Theorem 6.22. Let f : V → V be a rigid motion on a finite-dimensional real inner product space ...

height of a paralleogram

That's great, because you just learned how to find the height of a parallelogram. According to the eraser's original packaging, the area of the parallelogram is 12 square centimeters and the base is 6 centimeters. Let's plug 12 and 6 into the formula.

base of a paralleogram

The Base of a Parallelogram. A base of a parallelogram is defined to be any one of the sides of the parallelogram. There are two possible values for the height of the parallelogram, depending on which side of the parallelogram is chosen as the base.

altitudes of a paralleogram

The altitude (or height) of a parallelogram is the perpendicular distance from the base to the opposite side (which may have to be extended). In the figure above, the altitude corresponding to the base CD is shown. Opposite sides are congruent (equal in length) and parallel.

theorem 6.20

The area of a region is the sum of the areas of its nonoverlapping parts. Area of a Rectangle (Theorem 6.20). The area of a rectangle is the product of its base and height. Area of a Parallelogram (Theorem 6.21). The area of a parallelogram is the product of a base and its corresponding height. Area of a Triangle (Theorem ...

consecutive angles

The pairs of angles on one side of the transversal but inside the two lines are called Consecutive Interior Angles.

radius of are regular polygon

The radius of a regular polygon is the distance from the center to any vertex. It will be the same for any vertex. The radius is also the radius of the polygon's circumcircle, which is the circle that passes through every vertex. In this role, it is sometimes called the circumradius.

opposites angles of quadrilteral

The sum of the opposite angles of a "simple" quadrilateral in a circle is 180°. Drag the vertices on the circle to change the angles.

theorem 10.11

Theorem 10.11 Class 9 - The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degree ...

theorem 10.5

Theorem 10.5 : There is one and only one circle passing through three given non collinear points. ... THEOREM ...

theorem 6.10

Theorem 6.10: (ASA) Two triangles are congruent if and only if two angles and the side between them in one triangle are congruent to two angles and the side between them in a second triangle, then the triangles are congruent. Proof: If two angles in one triangle are congruent to two corresponding angles in a second ...

theorem 6.13

Theorem 6.11. A parallelogram is a rhombus if and only its diagonals are perpendicular. Theorem 6.12. A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. Theorem 6.13. A parallelogram is a rectangle if and only if its diagonals are congruent. Theorem 6.14. If a trapezoid is isosceles, ...

theorem 6.11

Theorem 6.11: (AAS) Two triangles are congruent if and only if two angles and the side next to one of them in one triangle are congruent to the corresponding two angles and a side in a second triangle, then the triangles are congruent. Proof: If two angles in one triangle are congruent to two corresponding angles in a ...

theorem 6.12

Theorem 6.12: (HS) Two right triangles are congruent if and only if their hypoteneuses and one other side are congruent. Proof: If their hypoteneuses and one other side are congruent, then one could use the Pythagorean Theorem, Theorem 3.4 to conclude that the other sides are also congruent, so the triangles are ...

theorem 6.17

Theorem 6.14. If a trapezoid is isosceles, then each pair of base angles is congruent. Theorem 6.15. If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. Theorem 6.16. A trapezoid is isosceles if and only if its diagonals are congruent. Theorem 6.17. The midsegment of a trapezoid is parallel to ...

theorem 6.19

Theorem 6.19: SAA Congruence Theorem: If two angles of a triangle and a side opposite one of the two angles are congruent to the corresponding angles and side of another triangle, then the two triangles are congruent. Proof : Given: ∠A ≅ ∠X; ∠C ≅ ∠Z; AB ≅ XY. Prove: ∆ABC ≅ ∆XYZ. Statement. Reason. 1. 1. 2. 2. 3.

theorem 6.5

Theorem 6.5: Two line segments are congruent if and only if they have the same length. Proof: If the segments are congruent, then it is possible to move one onto the other by an isometry. Since isometries preserve distances, the distance between the endpoints, which is the same thing as the lengths, are the same in both ...

theorem 6.8

Theorem 6.8: (SAS) Two triangles are congruent if and only if two sides and the angle between them in one triangle are congruent to the two sides and the angle between them in the other triangle. Proof: If the two triangles are congruent, then when the whole triangle is moved to the other triangle, the three angles and three ...

theorem 6.9

Theorem 6.9: If triangles are similar then the ratios the lengths of their corresponding sides are all the same. Proof: We show that if the angles are congruent that the ratios of the lengths of the corresponding sides are all the same. In the illustration ...

area of a triangles

To find the area of a triangle, multiply the base by the height, and then divide by 2. The division by 2 comes from the fact that a parallelogram can be divided into 2 triangles. For example, in the diagram to the left, the area of each triangle is equal to one-half the area of the parallelogram.

height of a trapezoid

To solve this question, you must divide the trapezoid into a rectangle and two right triangles. Using the Pythagorean Theorem, you would calculate the height of the triangle which is 4. The dimensions of the rectangle are 5 and 4, hence the area will be 20.

theorem 6.6

Translate A' to A. Rotate the plane about A until B' is on the ray from A through B. By Theorem 5.13, there are two angles having the ray from A through B as one arm, and they are both reflections of each other about the line determined by AB. So if B' is not on the ray from A through B, then it will be after reflecting about the ...

height of a triangle

You can use any one altitude-base pair to find the area of the triangle, via the formula A = (1/2)bh. In each of those diagrams, the triangle ABC is the same. The green line is the altitude, the "height", and the side with the red perpendicular square on it is the "base." All three sides of the triangle get a turn.Mar 18, 2013

quadrilateral

a four-sided figure.

parallelogram

a four-sided plane rectilinear figure with opposite sides parallel.

apothem

a line from the center of a regular polygon at right angles to any of its sides.

rhombus

a parallelogram with opposite equal acute angles, opposite equal obtuse angles, and four equal sides.

polygon

a plane figure with at least three straight sides and angles, and typically five or more.

octagon

a plane figure with eight straight sides and eight angles.

pentagon

a plane figure with five straight sides and five angles.

square

a plane figure with four equal straight sides and four right angles.

rectangle

a plane figure with four straight sides and four right angles, especially one with unequal adjacent sides, in contrast to a square.

nonagon

a plane figure with nine straight sides and nine angles.

heptagon

a plane figure with six straight sides and angles.

decagon

a plane figure with ten straight sides and angles.

triangle

a plane figure with three straight sides and three angles.

dodecagon

a plane figure with twelve sides.

trapezoid

a quadrilateral with only one pair of parallel sides.

regular polygon

a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length)

kite

a toy consisting of a light frame with thin material stretched over it, flown in the wind at the end of a long string.

isosceles trapezoid

is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides.

equiangular polygon

is a polygon whose vertex angles are equal. If the lengths of the sides are also equal

convex polygon

is defined as a polygon with all its interior angles less than 180°.

concave polygon

is defined as a polygon with one or more interior angles greater than 180°.

equilateral polygon

make a polygon and an equilateral polygon is a polygon which has all sides of the same length

theorem 6.14

theorem 6.13. a parallelogram is arectangle if and only if its diagonals are conguent. theorem 6.14. if a trapezoid is isosceles then each pair of base angles is conguent. theorem 6.15. if a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. theorem 6.16. a trapezoid is isosceles if and only if its ...

theorem 6.15

theorem 6.13. a parallelogram is arectangle if and only if its diagonals are conguent. theorem 6.14. if a trapezoid is isosceles then each pair of base angles is conguent. theorem 6.15. if a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. theorem 6.16. a trapezoid is isosceles if and only if its ...

theorem 6.16

theorem 6.13. a parallelogram is arectangle if and only if its diagonals are conguent. theorem 6.14. if a trapezoid is isosceles then each pair of base angles is conguent. theorem 6.15. if a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. theorem 6.16. a trapezoid is isosceles if and only if its ...