Exercise 132 Page 524

a To prove congruence between two triangles, we need to know at least one of the following.

If there are three pairs of congruent sides.
Two pairs of sides and the included angle are congruent.
There are at least two pairs of congruent angles and one pair of congruent corresponding sides.
Examining the diagram we see that △ ABD and △ CBD share BD as a side, which means it is congruent according to the Reflexive Property of Congruence.

With this information, we can claim congruence by the SAS (Side-Angle-Side) Congruence Theorem.

Side-Angle-Side Congruence Theorem

If two sides and the included angle in a triangle are congruent to corresponding parts in another triangle, then the triangles are congruent.

We write it as △ ABD ≅ △ CBD.

Side-Angle-Side Congruence Theorem
If two sides and the included angle in a triangle are congruent to corresponding parts in another triangle, then the triangles are congruent.

b Like in Part A, we do not have any expressions for the triangle's sides. Therefore, we must first prove that they have at least two pairs of congruent angles to first claim similarity. Examining the diagram, we can identify a pair of vertical angles which are congruent according to the Vertical Angles Theorem.

If we view AD and BC as a transversal, we can claim that ∠ A≅ ∠ D and that ∠ B ≅ ∠ C by the Alternate Interior Angles Theorem.

The angles have three pairs of congruent angles, which means we know they are similar triangles. However, to claim congruence we need to know that at least one pair of corresponding sides is congruent. Since we do not have this information, we cannot claim congruence.

c In both triangles we know two angles. Using the Triangle Angle Sum Theorem, we can determine the last angle in each triangle.

m∠ D+45^(∘) +62^(∘) = 180^(∘) ⇔ m∠ D=73^(∘) m∠ B+73^(∘) +62^(∘) = 180^(∘) ⇔ m∠ B=45^(∘)

The angles have three pairs of congruent angles, which means we can claim that they are similar triangles. To investigate if they are congruent we have to identify corresponding sides, which are sides that are between the same two pairs of congruent angles.

Since the side that is 13 is the included side to the same two pairs of congruent angles, we know that these triangles are congruent by the ASA (Angle-Side-Angle) Congruence Theorem.

Angle-Side-Angle Congruence Theorem
If two angles and the included side of a triangle are congruent with two angles and their included side of another triangle, then the triangles are congruent.

We write it as △ ABC ≅ △ DEF.

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Exercise 132 Page 524

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