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# 4.6: Chapter 4 Review

Difficulty Level: At Grade Created by: CK-12

Definitions, Postulates, and Theorems

Interior Angles
The angles inside of a closed figure with straight sides.
Vertex
The point where the sides of a polygon meet.
Triangle Sum Theorem
The interior angles of a triangle add up to 180\begin{align*}180^\circ\end{align*}.
Exterior Angle
The angle formed by one side of a polygon and the extension of the adjacent side.
Exterior Angle Sum Theorem
Each set of exterior angles of a polygon add up to 360\begin{align*}360^\circ\end{align*}.
Remote Interior Angles
The two angles in a triangle that are not adjacent to the indicated exterior angle.
Exterior Angle Theorem
The sum of the remote interior angles is equal to the non-adjacent exterior angle.
Congruent Triangles
Two triangles are congruent if the three corresponding angles and sides are congruent.
Third Angle Theorem
If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles must also congruent.
Reflexive Property of Congruence
Any shape is congruent to itself.
Symmetric Property of Congruence
If two shapes are congruent, the statement can be written with either shape on either side of the \begin{align*}\cong\end{align*} sign.
Transitive Property of Congruence
If two shapes are congruent and one of those is congruent to a third, the first and third shapes are also congruent.
Side-Side-Side (SSS) Triangle Congruence Postulate
If three sides in one triangle are congruent to three sides in another triangle, then the triangles are congruent.
Included Angle
When an angle is between two given sides of a triangle (or polygon).
Side-Angle-Side (SAS) Triangle Congruence Postulate
If two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle, then the two triangles are congruent.
Angle-Side-Angle (ASA) Congruence Postulate
If two angles and the included side in one triangle are congruent to two angles and the included side in another triangle, then the two triangles are congruent.
Angle-Angle-Side (AAS or SAA) Congruence Theorem
If two angles and a non-included side in one triangle are congruent to two corresponding angles and a non-included side in another triangle, then the triangles are congruent.
HL Congruence Theorem
If the hypotenuse and leg in one right triangle are congruent to the hypotenuse and leg in another right triangle, then the two triangles are congruent.
Base Angles Theorem
The base angles of an isosceles triangle are congruent.
Isosceles Triangle Theorem
The angle bisector of the vertex angle in an isosceles triangle is also the perpendicular bisector to the base.
Base Angles Theorem Converse
If two angles in a triangle are congruent, then the opposite sides are also congruent.
Isosceles Triangle Theorem Converse
The perpendicular bisector of the base of an isosceles triangle is also the angle bisector of the vertex angle.
Equilateral Triangles Theorem
all sides in an equilateral triangle have exactly the same length.

## Review

For each pair of triangles, write what needs to be congruent in order for the triangles to be congruent. Then, write the congruence statement for the triangles.

1. HL
2. ASA
3. AAS
4. SSS
5. SAS

Using the pictures below, determine which theorem, postulate or definition that supports each statement below.

1. m1+m2=180\begin{align*}m \angle 1+m \angle 2=180^\circ\end{align*}
2. 56\begin{align*}\angle 5 \cong \angle 6\end{align*}
3. m1=m4+m3\begin{align*}m \angle 1=m \angle 4+ m \angle 3\end{align*}
4. m8=60\begin{align*}m \angle 8 = 60^\circ\end{align*}
5. m5+m6+m7=180\begin{align*}m \angle 5+m \angle 6+m \angle 7=180^\circ\end{align*}
6. 8910\begin{align*}\angle 8 \cong \angle 9 \cong \angle 10\end{align*}
7. If m7=90\begin{align*}m \angle 7 = 90^\circ\end{align*}, then m5=m6=45\begin{align*}m \angle 5 = m \angle 6 = 45^\circ\end{align*}

## Texas Instruments Resources

In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9689.

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