Question

solve for x.
solve for x and y.
solve for x and y.
two angles in a triangle measure 27° and 83°. what is the measure of the third angle?
a 110°
b 70°
c 80°
d cannot be determined.

Explanation:

Step1: Sum of angles in a triangle

The sum of angles in a triangle is \(180^{\circ}\).

Step2: Solve for \(x\) in the first triangle

Given two angles \(64^{\circ}\) and \(64^{\circ}\), then \(x = 180-(64 + 64)\)

$$x=180 - 128=52$$

Step3: Solve for \(x\) in the second triangle (using vertical angles and sum of angles)

First, find the third angle in the left - hand triangle: \(180-(42 + 41)=97^{\circ}\). Since \(x\) is a vertical angle to \(97^{\circ}\), \(x = 97\). Then for \(y\), using sum of angles in the right - hand triangle: \(y=180-(97 + 46)=37\)

Step4: Solve for \(x\) and \(y\) in the third triangle (isosceles triangle properties)

Since the two sides are equal, the base angles are equal. Let the base angles be \(x\). Then \(2x+72 = 180\), \(2x=108\), \(x = 54\). And \(y=180 - 54=126\)

Step5: Solve the multiple - choice question

Let the third angle be \(z\). Using the sum of angles in a triangle formula \(z=180-(27 + 83)\)

$$z=180 - 110=70$$

Answer:

1. \(x = 52\)
2. \(x = 97,y = 37\)
3. \(x = 54,y = 126\)
4. B. \(70^{\circ}\)