a What type of angle pair does ∠ABD and ∠EBC make?
B
b Corresponding sides are between the same pair of corresponding angles.
A
a They are similar. See solution.
B
b They are congruent. See solution.
Practice makes perfect
a If the given triangles are similar they will have at least two pairs of congruent angles. Examining the diagram, we see that they have one pair of congruent angles, ∠BAD≅ ∠BEC. Also, from the diagram we can identify a straight angle pair.
The angles of a straight angle pair are supplementary. Since one angle is 90^(∘), the second angle must also be 90^(∘).
Now we see that the triangles have two pairs of congruent angles, which means we know they are similar by the AA~ condition. Let's show this proof as a flowchart.
b If the triangles are congruent they will, in addition to having the same shape, also have the same size. If we can show that at least one pair of corresponding sides have the same shape, we know this is the true. Corresponding sides are between the same two pairs of corresponding angles. With this information we can identify corresponding sides.
As we can see, BD and BC are corresponding sides. Since we know the hypotenuse and a leg in â–³ ABD, we can calculate the unknown leg, BD, by using the Pythagorean Theorem. If it turns out to be 9 units, we know that the triangles are congruent.
