a Is △ ABD a right triangle? How can we show that?
B
b Is △ CEG a right triangle? How can we show that?
A
a See solution.
B
b Yes
Explanation: See solution.
Practice makes perfect
a Let's highlight △ ABD and △ BCD in our diagram.
In the green triangle, we see from the marked angle that it's a right triangle. We also know that one of it's legs and hypotenuse are congruent with two sides in the blue triangle. We refer to the blue triangles sides as "sides", because we first have to prove that it's a right triangle.
Proving △ ABD is a Right Triangle
If we examine ∠ ADB, we see that it forms a linear pair with ∠ CDB. According to the Linear Pair Postulate, they are supplementary angles which means their angle measures add up to 180^(∘). Since ∠ CDB is a right angle, it must be that ∠ ADB is a right angle as well.
Now we have enough information to, by the HL Congruence Theorem, claim that the two triangles are congruent.
b By the same reasoning as in Part A, we can claim that △ CEG is a right triangle as well.
Again, using the HL Congruence Theorem, we can prove that all four triangles are congruent.
