Question

solve for x and y.
solve for x, y and z.
solve for x.
solve for x.

Explanation:

Step1: Sum of angles in a triangle

The sum of angles in a triangle is \(180^{\circ}\). For the first triangle in 1: \(64 + 84+x=180\), so \(x = 180-(64 + 84)=32\). For the second triangle in 1: \(56+78 + y=180\), so \(y=180-(56 + 78)=46\).

Step2: Exterior - angle theorem and angle - sum property

In 2: Using the exterior - angle theorem, \(x=35 + 36=71\). Then, for the left - hand triangle, \(z + 50+(y)=180\). Since \(y = 180 - x=180 - 71 = 109\) (linear pair), for the left - hand triangle \(z+50 + 109=180\), so \(z=180-(50 + 109)=21\).

Step3: Exterior - angle theorem

In 3: By the exterior - angle theorem, \(8x-4=50+(4x + 6)\).

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Step4: Sum of angles in a right - triangle

In 4: The sum of angles in a right - triangle (\(90^{\circ}\) angle) is \(180^{\circ}\). So \(90+34+(2x + 6)=180\).

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Answer:

1. \(x = 32,y = 46\); 2. \(x = 71,y = 109,z = 21\); 3. \(x = 15\); 4. \(x = 25\)