Question

i an isosceles triangle has base angles that each measure 38°. find x, the measure of the third angle of this isosceles triangle in degrees.
u the angle measures of a right - triangle are shown in the diagram. what is the value of y?
c the angles in a triangle have measures of 61°, (3z)°, (z - 5)°. what is the value of z?
p the angle measures of an isosceles triangle are shown. find the value of w.

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°.

Step2: Solve for x in the first isosceles triangle

We have two base angles of 38° each. Let the third angle be x. Then, using the angle - sum property, we get the equation \(x + 38^{\circ}+38^{\circ}=180^{\circ}\). Simplifying, \(x=180^{\circ}-(38^{\circ} + 38^{\circ})=180^{\circ}-76^{\circ}=104^{\circ}\).

Step3: Solve for y in the right - triangle

In a right - triangle, one angle is 90°. The other non - right angle is 24°. Let the third angle be \(2y\). Then, \(90^{\circ}+24^{\circ}+2y^{\circ}=180^{\circ}\). First, combine like terms: \(114^{\circ}+2y^{\circ}=180^{\circ}\). Then, subtract 114° from both sides: \(2y^{\circ}=180^{\circ}-114^{\circ}=66^{\circ}\). Divide both sides by 2: \(y = 33\).

Step4: Solve for z in the third triangle

The angles of the triangle are 61°, \(3z^{\circ}\), and \((z - 5)^{\circ}\). Using the angle - sum property, \(61^{\circ}+3z^{\circ}+(z - 5)^{\circ}=180^{\circ}\). Combine like terms: \(61^{\circ}-5^{\circ}+3z^{\circ}+z^{\circ}=180^{\circ}\), which gives \(56^{\circ}+4z^{\circ}=180^{\circ}\). Subtract 56° from both sides: \(4z^{\circ}=180^{\circ}-56^{\circ}=124^{\circ}\). Divide both sides by 4: \(z = 31\).

Step5: Solve for w in the fourth isosceles triangle

In an isosceles triangle, two angles are equal. Let's assume the equal angles are \(8w^{\circ}\) and \(9w - 1^{\circ}\). Case 1: If \(8w=9w - 1\), then \(9w-8w = 1\), so \(w = 1\). But if we check the angle - sum property with \(w = 1\), the angles are \(8^{\circ}\), \(8^{\circ}\), and \(62^{\circ}\), and \(8^{\circ}+8^{\circ}+62^{\circ}
eq180^{\circ}\). So, the equal angles must be \(8w^{\circ}\) and \(8w^{\circ}\). Then, \(8w+8w+62^{\circ}=180^{\circ}\). Combine like terms: \(16w^{\circ}=180^{\circ}-62^{\circ}=118^{\circ}\), \(w=\frac{118^{\circ}}{16}=7.375\). Or, if we assume the equal angles are \(9w - 1^{\circ}\) and \(9w - 1^{\circ}\), then \((9w - 1)+(9w - 1)+62^{\circ}=180^{\circ}\). Combine like terms: \(18w-2 + 62^{\circ}=180^{\circ}\), \(18w+60^{\circ}=180^{\circ}\), \(18w=120^{\circ}\), \(w=\frac{20}{3}\approx6.67\). If we use the fact that the sum of angles in a triangle is 180° and assume the non - equal angle is 62°, and the equal angles are \(8w\) and \(9w - 1\), we set up the equation \(8w+(9w - 1)+62=180\), \(17w+61 = 180\), \(17w=119\), \(w = 7\).

Answer:

x = 104, y = 33, z = 31, w = 7