Question

period:______ geometry homework 2.03: swbat solve for missing angles using special angle relationships created when 2 parallel lines are cut by a transversal show all work and annotations for full credit. #1.) in the diagram below \\(\overline{ab}\parallel\overline{cd}\\). if the \\(m\angle3=(4x + 29)^{circ}\\) and the \\(m\angle2=(x + 41)^{circ}\\) determine \\(m\angle1\\). what angle pair and relationship allowed you to solve for \\(x\\)? #2.) in the diagram below, \\(\overline{ab}\\) and \\(\overline{cd}\\) are parallel and \\(m\angle7=(x^{2}+90)^{circ}\\) and \\(m\angle2=(35 - 6x)^{circ}\\). determine \\(m\angle5\\).

Explanation:

Step1: Identify angle - pair relationship

Since $\overleftrightarrow{AB}\parallel\overleftrightarrow{CD}$, $\angle2$ and $\angle3$ are alternate - interior angles. Alternate - interior angles are congruent when two parallel lines are cut by a transversal. So, $m\angle2 = m\angle3$.
$(x + 41)=(4x+29)$

Step2: Solve for x

Subtract $x$ from both sides:
$41 = 3x+29$
Subtract 29 from both sides:
$3x=41 - 29=12$
Divide both sides by 3:
$x = 4$

Step3: Find $m\angle1$

$\angle1$ and $\angle2$ are a linear pair. A linear pair of angles is supplementary, so $m\angle1+m\angle2 = 180^{\circ}$.
First, find $m\angle2$: $m\angle2=(x + 41)^{\circ}$, substituting $x = 4$, we get $m\angle2=(4 + 41)^{\circ}=45^{\circ}$.
Then, $m\angle1=180 - m\angle2=180 - 45=135^{\circ}$

Step1: Identify angle - pair relationship

Since $\overleftrightarrow{AB}\parallel\overleftrightarrow{CD}$, $\angle2$ and $\angle7$ are same - side interior angles. Same - side interior angles are supplementary when two parallel lines are cut by a transversal. So, $m\angle2+m\angle7 = 180^{\circ}$.
$(35-6x)+(x^{2}+90)=180$
$x^{2}-6x + 125=180$
$x^{2}-6x - 55=0$

Step2: Solve the quadratic equation

Factor the quadratic equation: $x^{2}-6x - 55=(x - 11)(x + 5)=0$
So, $x=11$ or $x=-5$

Step3: Find $m\angle5$

$\angle5$ and $\angle7$ are vertical angles, so $m\angle5=m\angle7$.
Case 1: If $x = 11$, $m\angle7=(x^{2}+90)=(11^{2}+90)=(121 + 90)=211^{\circ}$ (not valid as angle measure must be between $0^{\circ}$ and $180^{\circ}$)
Case 2: If $x=-5$, $m\angle7=((-5)^{2}+90)=(25 + 90)=115^{\circ}$
So, $m\angle5 = 115^{\circ}$

Answer:

$m\angle1 = 135^{\circ}$

For the second - part of the question: