Question

1. given: lines x and y are parallel. using your understanding of angle - theorems and substitution, prove that < h = < d. 4. explain why a + b = d.

Explanation:

Step1: Identify corresponding angles

Since lines x and y are parallel, $\angle h$ and $\angle f$ are corresponding angles, so $\angle h=\angle f$ (Corresponding - angles postulate).

Step2: Identify vertical - angles

$\angle f$ and $\angle d$ are vertical angles, so $\angle f = \angle d$ (Vertical - angles are congruent).

Step3: Use substitution

By substitution, since $\angle h=\angle f$ and $\angle f = \angle d$, then $\angle h=\angle d$.

for second part:

Step1: Use the exterior - angle theorem

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles. In $\triangle ABC$, $\angle d$ is an exterior angle and $\angle a$ and $\angle b$ are the non - adjacent interior angles. So, by the exterior - angle theorem of a triangle, $a + b=d$.

Answer:

Statement	Reason
$\angle h=\angle f$	Corresponding - angles postulate
$\angle f = \angle d$	Vertical - angles are congruent
$\angle h=\angle d$	Substitution