Question

understand
angle relationships in triangles
name:
prerequisite: how can you use what you know
about angle relationships to solve problems?
study the example showing how to find the measure of a
missing angle. then solve problems 1-6.
example
$overline{ab}$ and $overline{ad}$ of parallelogram abcd are extended as
shown. find the measure of $\angle fab$.
$\angle fab$ and $\angle fae$ are supplementary angles, so the
sum of the measures of $\angle fab$ and $\angle fae$ is $180^\circ$.
$(x + 7) + (2x - 1) = 180$
$x + 7 + 2x - 1 = 180$
$3x + 6 = 180$
$3x = 174$
$x = 58$
this means the measure of $\angle fab$ is $(2x - 1)^\circ =
(2(58) - 1)^\circ = 115^\circ$.
1 what is $m\angle fae$? explain your reasoning.
2 what is $m\angle bad$? explain your reasoning.
3 explain a way to find $m\angle abc$. then find $m\angle abc$.
lesson 22 understand angle relationships in trian

Explanation:

Step1: Use known x value for ∠FAE

From the example, $x=58$. Substitute into $m\angle FAE=(x+7)^\circ$.
$m\angle FAE=(58+7)^\circ=65^\circ$

Step2: Identify ∠BAD and ∠FAE

∠BAD and ∠FAE are vertical angles, so they are congruent.
$m\angle BAD=m\angle FAE=65^\circ$

Step3: Use parallelogram consecutive angles

In parallelogram ABCD, ∠BAD and ∠ABC are supplementary.
$m\angle ABC=180^\circ - m\angle BAD=180^\circ-65^\circ=115^\circ$

Answer:

1. $m\angle FAE=65^\circ$. We substitute the solved $x=58$ into the expression for $\angle FAE$, which is $(x+7)^\circ$.
2. $m\angle BAD=65^\circ$. $\angle BAD$ and $\angle FAE$ are vertical angles, so their measures are equal.
3. In parallelogram $ABCD$, consecutive angles are supplementary, so subtract $m\angle BAD$ from $180^\circ$. $m\angle ABC=115^\circ$