Question

unit 4 quiz: triangles
version 1
name: thiago goncalcz
solve for the given variable. write your answer on the line provided. show
all of your work! round all answers to the hundredth place.

1. solve for t.
2. solve for m∠c.

Explanation:

1. Solve for \( t \)

Step1: Identify the triangle type

Since \(\angle G = 45^{\circ}\) and \(\angle I=45^{\circ}\), \(\triangle GHI\) is a \(45 - 45-90\) triangle. In a \(45 - 45 - 90\) triangle, the legs are equal. So \(GH = HI\).

Step2: Set up the equation

We have \(3t - 7=t + 1\).
Subtract \(t\) from both sides: \(3t-t-7=t - t+ 1\), which simplifies to \(2t-7 = 1\).
Add 7 to both sides: \(2t-7 + 7=1 + 7\), so \(2t=8\).
Divide both sides by 2: \(t=\frac{8}{2}=4\).

2. Solve for \(m\angle C\)

Step1: Identify the triangle type

Since \(AB = BC\) (marked as equal), \(\triangle ABC\) is an isosceles triangle. In an isosceles triangle, the base - angles are equal. So \(\angle B=\angle A = 63^{\circ}\).

Step2: Use the angle - sum property of a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\). Let \(m\angle C=x\). Then \(x+\angle A+\angle B = 180^{\circ}\).
Substitute \(\angle A = 63^{\circ}\) and \(\angle B = 63^{\circ}\) into the equation: \(x+63^{\circ}+63^{\circ}=180^{\circ}\).
Simplify the left - hand side: \(x + 126^{\circ}=180^{\circ}\).
Subtract \(126^{\circ}\) from both sides: \(x=180^{\circ}-126^{\circ}=54^{\circ}\).

Answer:

1. \(t = 4\)
2. \(m\angle C=54^{\circ}\)