Question

because two congruent angles.

1. if $\triangle abc$ and $\triangle edc$ are similar, what is the value of $x$?
2. is $\triangle xyz \sim \triangle ghi$? explain.
3. construct arguments is $\triangle qrt \sim \triangle ghi$? explain.

Explanation:

For Question 2:

Step1: Identify vertical angles

Vertical angles $\angle ACB$ and $\angle DCE$ are equal:
$4x - 8 = 3x + 15$

Step2: Solve for x

Subtract $3x$ from both sides:
$4x - 3x - 8 = 15$
$x - 8 = 15$
Add 8 to both sides:
$x = 15 + 8$

First, calculate the missing angle in $\triangle XYZ$: $180^\circ - 45^\circ - 31^\circ = 104^\circ$. $\triangle GHI$ has angles $45^\circ$, $104^\circ$, and $31^\circ$ (since $180-45-104=31$). All corresponding angles of the two triangles are congruent, so by AA (Angle-Angle) similarity criterion, the triangles are similar.

First, find the missing angle in $\triangle QRT$: $\angle QRT = 54^\circ$, $\angle QTR = 31^\circ$, so $\angle RQT = 180^\circ - 54^\circ - 31^\circ = 95^\circ$. In $\triangle GHI$, $\angle GHI = 95^\circ$, $\angle GIH = 31^\circ$, so $\angle HGI = 180^\circ - 95^\circ - 31^\circ = 54^\circ$. All corresponding angles of $\triangle QRT$ and $\triangle GHI$ are congruent, so they meet the AA similarity criterion.

Answer:

$x = 23$

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For Question 3: