CK-12 Geometry Concepts

http://www.ck12.org Chapter 4. Triangles and Congruence

CK-12 Foundation: Chapter4CongruentTrianglesB

Concept Problem Revisited

There are 16 “A” triangles and they are all congruent. There are 16 “B” triangles and they are all congruent. The
quilt pattern is made from dividing up the square into smaller squares. The “A” triangles are all 321 of the overall
square and the “B” triangles are each 1281 of the large square. Both the “A” and “B” triangles are right triangles.

Vocabulary

Two figures arecongruentif they have exactly the same size and shape. Two triangles arecongruentif their three
pairs of corresponding angles and three pairs of corresponding sides are congruent.

Guided Practice

1. Determine if the triangles are congruent using the definition of congruent triangles.


2. Determine if the triangles are congruent using the definition of congruent triangles.


3. Determine if the triangles are congruent using the definition of congruent triangles.

Answers:

1. We can see from the markings that^6 B∼=^6 C,^6 A∼=^6 D, and^6 AEB∼=^6 DECbecause they are vertical angles.
   Also, we know thatBA∼=CD,EA∼=ED, andBE∼=CE. Because three pairs of sides and three pairs of angles are
   all congruent and they are corresponding parts, this means that the two triangles are congruent.


2. While there are congruent corresponding parts, there are only two pairs of congruent sides, the marked ones and
   the shared side. Without knowing whether or not the third pair of sides is congruent we cannot say if the triangles
   are congruent using the definition of congruent triangles.


3. We can see from the markings that^6 G∼=^6 L,^6 F∼=^6 K, and therefore^6 H∼=^6 Mby the Third Angle Theorem.
   Also, we know thatMK∼=F H,GF∼=LK, andGH∼=LM. Because three pairs of sides and three pairs of angles are
   all congruent and they are corresponding parts, this means that the two triangles are congruent.

Practice

The following illustrations show two parallel lines cut by a transversal. Are the triangles formed definitively
congruent?

1.

2.

3.

4.

5.

Based on the following details, are the triangles definitively congruent?

6. Both triangles are right triangles in which one angle measures 55◦. All of their corresponding sides are
   congruent.


7. Both triangles are equiangular triangles.


8. Both triangles are equilateral triangles. All sides are 5 inches in length.