Question

1. in the diagram to the right $overline{ah}$ and $overline{cd}$ intersect at e. a) find the measure of $angle hec$ when $mangle aed = 65$. b) name the geometric relationship between $angle hec$ and $angle aed$. c) find the measure of $angle deh$. 2) $overline{mn}$ and $overline{rs}$ intersect at t. if $mangle rtm=5x$ and $mangle nts = 3x + 10$, find $mangle rtm$. (include a diagram) 3) given $mangle abc = 4x$ and $mangle cbd = 60$. find the value of $x$ and $mangle ebd$

Explanation:

1.

a)

Step1: Identify vertical - angle property

Vertical angles are equal. $\angle HEC$ and $\angle AED$ are vertical angles.

Step2: Determine angle measure

Since $\angle HEC=\angle AED$ and $m\angle AED = 65^{\circ}$, then $m\angle HEC=65^{\circ}$.

Vertical angles are formed when two lines intersect. $\angle HEC$ and $\angle AED$ are formed by the intersection of $\overline{AH}$ and $\overline{CD}$.

Step1: Recall linear - pair property

$\angle AED$ and $\angle DEH$ form a linear - pair. The sum of angles in a linear - pair is $180^{\circ}$.

Step2: Calculate $\angle DEH$

$m\angle DEH=180 - m\angle AED$. Since $m\angle AED = 65^{\circ}$, then $m\angle DEH=180 - 65=115^{\circ}$.

Answer:

$65^{\circ}$

b)