Question

name: jazzya k
period:
date:
seat:
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 $overline{ab}paralleloverline{cd}$. if the $mangle3=(4x + 29)^{circ}$ and the $mangle2=(x + 41)^{circ}$ determine $mangle1$.
what angle pair and relationship allowed you to solve for $x$?
**#2.) in the diagram below, $overline{ab}$ and $overline{cd}$ are parallel and $mangle7=(x^{2}+90)^{circ}$ and $mangle2=(35 - 6x)^{circ}$. determine $mangle5$.
b. what angle pair and relationship allowed you to solve for $x$?
what angle pair and relationship allowed you to solve for $angle5$?

Explanation:

Step1: Identify angle - pair relationship for #1

$\angle2$ and $\angle3$ are alternate - interior angles. Since $\overline{AB}\parallel\overline{CD}$, alternate - interior angles are congruent. So, $m\angle2 = m\angle3$.
$x + 41=4x + 29$

Step2: Solve the equation for $x$ in #1

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

Step3: Find $m\angle2$

Substitute $x = 4$ into the expression for $m\angle2$: $m\angle2=(x + 41)^{\circ}=(4 + 41)^{\circ}=45^{\circ}$.

Step4: Find $m\angle1$

$\angle1$ and $\angle2$ are a linear - pair. A linear - pair of angles is supplementary, so $m\angle1=180 - m\angle2$. Then $m\angle1=180 - 45=135^{\circ}$.

Step5: Identify angle - pair relationship for #2

$\angle2$ and $\angle7$ are same - side interior angles. Since $\overline{AB}\parallel\overline{CD}$, same - side interior angles are supplementary. So, $m\angle2+m\angle7 = 180$.
$(35-6x)+(x^{2}+90)=180$

Step6: Solve the equation for $x$ in #2

Combine like terms: $x^{2}-6x + 125=180$. Rearrange to get a quadratic equation: $x^{2}-6x - 55 = 0$. Factor the quadratic: $(x - 11)(x+5)=0$. So, $x = 11$ or $x=-5$.

Step7: Find $m\angle2$ (check valid $x$ values)

If $x = 11$, $m\angle2=35-6\times11=35 - 66=-31$ (not valid as angle measure cannot be negative). If $x=-5$, $m\angle2=35-6\times(-5)=35 + 30 = 65^{\circ}$.

Step8: Find $m\angle5$

$\angle2$ and $\angle5$ are corresponding angles. Since $\overline{AB}\parallel\overline{CD}$, corresponding angles are congruent. So, $m\angle5=m\angle2 = 65^{\circ}$.

Answer:

1. $m\angle1 = 135^{\circ}$; Angle - pair: Alternate - interior angles ($\angle2$ and $\angle3$) were used to solve for $x$.
2. $m\angle5 = 65^{\circ}$; Angle - pair: Same - side interior angles ($\angle2$ and $\angle7$) were used to solve for $x$, and corresponding angles ($\angle2$ and $\angle5$) were used to solve for $\angle5$.