Question

1. ∠adc = 110°. find ∠bdc =, ∠ade =, ∠bde =. write down pairs of vertical angles. write down pairs of supplementary angles. 4) given ab || cd, ∠aeg = 105°. find ∠geb, ∠aef, ∠bef, ∠cfe, ∠dfe, ∠cfh, ∠dfh. write down pairs of vertical angles.

Explanation:

Step1: Solve for $\angle BDC$ in problem 3

Since $\angle ADC = 110^{\circ}$ and $\angle ADC+\angle BDC = 180^{\circ}$ (linear - pair of angles), then $\angle BDC=180^{\circ}-\angle ADC$.
$\angle BDC = 180 - 110=70^{\circ}$

Step2: Solve for $\angle ADE$ in problem 3

$\angle ADE$ and $\angle BDC$ are vertical - angles. Vertical angles are equal. So $\angle ADE=\angle BDC = 70^{\circ}$

Step3: Solve for $\angle BDE$ in problem 3

$\angle BDE$ and $\angle ADC$ are vertical - angles. So $\angle BDE=\angle ADC = 110^{\circ}$

Step4: Identify vertical - angle pairs in problem 3

Vertical - angle pairs are: $\angle ADE$ and $\angle BDC$, $\angle BDE$ and $\angle ADC$

Step5: Identify supplementary - angle pairs in problem 3

Supplementary - angle pairs are: $\angle ADC$ and $\angle BDC$, $\angle ADE$ and $\angle BDE$

Step6: Solve for angles in problem 4

Since $\angle AEG = 105^{\circ}$, and $\angle AEG+\angle GEB=180^{\circ}$ (linear - pair of angles), then $\angle GEB = 180 - 105=75^{\circ}$
$\angle AEF$ and $\angle AEG$ are supplementary, so $\angle AEF = 180 - 105 = 75^{\circ}$
$\angle BEF=\angle AEG = 105^{\circ}$ (vertical - angles)
Since $AB\parallel CD$, $\angle CFE=\angle AEF = 75^{\circ}$ (alternate - interior angles)
$\angle DFE=\angle BEF = 105^{\circ}$ (alternate - interior angles)
$\angle CFH$ and $\angle DFE$ are vertical - angles, so $\angle CFH = 105^{\circ}$
$\angle DFH$ and $\angle CFE$ are vertical - angles, so $\angle DFH = 75^{\circ}$

Step7: Identify vertical - angle pairs in problem 4

Vertical - angle pairs are: $\angle AEF$ and $\angle BEF$, $\angle AEG$ and $\angle BEG$, $\angle CFE$ and $\angle DFH$, $\angle DFE$ and $\angle CFH$

Answer:

Problem 3:
$\angle BDC = 70^{\circ}$
$\angle ADE = 70^{\circ}$
$\angle BDE = 110^{\circ}$
Vertical - angle pairs: $\angle ADE$ and $\angle BDC$, $\angle BDE$ and $\angle ADC$
Supplementary - angle pairs: $\angle ADC$ and $\angle BDC$, $\angle ADE$ and $\angle BDE$
Problem 4:
$\angle GEB = 75^{\circ}$
$\angle AEF = 75^{\circ}$
$\angle BEF = 105^{\circ}$
$\angle CFE = 75^{\circ}$
$\angle DFE = 105^{\circ}$
$\angle CFH = 105^{\circ}$
$\angle DFH = 75^{\circ}$
Vertical - angle pairs: $\angle AEF$ and $\angle BEF$, $\angle AEG$ and $\angle BEG$, $\angle CFE$ and $\angle DFH$, $\angle DFE$ and $\angle CFH$