Exercise 75 Page 431

Practice makes perfect

a From the diagram, we see that the triangle has two pairs of congruent sides. We also see that the third side is shared which means this is congruent as well according to the Reflexive Property of Congruence.

With this information, we can claim congruence by the SSS Congruence condition.

b We see that the triangles have two pairs of congruent sides. We can also identify a pair of vertical angles. These angles are congruent by the Vertical Angles Theorem.

Knowing two pairs of congruent sides and a non-included angle is not enough information to determine congruence between the triangles.

c From the diagram, we see two pairs of congruent angles in our triangles. Also, notice that the triangles share part of their horizontal side which we will label x.

With this information, we can claim congruence by the ASA Congruence condition.

d In the bigger triangle, two sides are congruent which means this is an isosceles triangle. By the Base Angles Theorem, we know that the base angles of our triangle are congruent.

Note that the right angle and it's adjacent angle form a straight angle pair which means they have a sum of 180^(∘). Since one angle is 90^(∘), the second angle has to be 90^(∘) as well.