Answer number 34 and 35. For 34 find the measures of the numbered angles in each isosceles trapezoid. For number 35 find the measures of the numbered angles in each kite.

33. m∠1 = 135°, ∠3 = 45°, ∠2 = 135°

34. m∠1 = 80°, m∠2 = 100°, m∠3 = 100°

35. m∠1 = 90°, m∠2 = 25°

Step-by-step explanation:

33. The given parameters of the isosceles trapezoid are;

The measure of one of the base angles = 45°

For an isosceles trapezoid, the base angles are equal, therefore, we have;

The measure of the other base angle = m∠3 = 45°

Similarly, m∠1 = m∠2

Also, we have;

The sum of the interior angles in a trapezoid = The sum of the interior angles in a quadrilateral = 360°

∴ 45° + m∠3 + m∠1 + m∠2 = 360°

By substituting the known values, we have;

45° + 45° + m∠1 + m∠2 = 360°

From m∠1 = m∠2, we have;

45° + 45° + m∠1 + m∠1 = 90° + 2·m∠1 = 360°

2·m∠1 = 360° - 90° = 270°

∴ m∠1 = 270°/2 = 135°

m∠1 = 135° = m∠2

Therefore, we have;

∠3 = 45°, ∠1 = 135°, ∠2 = 135°

34. The measure of the base angles = 80°

∴ The measure of the other base angle = m∠1 = 80°

m∠2 = m∠3

80° + m∠1 + m∠2 + m∠3 = 360°

∴ 80° + 80° + 2·m∠2 = 360°

m∠2 = (360° - (80° + 80°))/2 = 100°

m∠2 = 100° = m∠3

35. The kite comprises of two half isosceles triangles

The base angles of the right half isosceles triangle = 65°

The angle formed by the diagonals of the kite = 90° (Perpendicular diagonals theorem)

Therefore, m∠1 = 90°

The diagonals bisect the angles at the vertices (Kite theorem)

Therefore, we have;

65° + 90° + m∠2 = 180°

m∠2 = 180° - (65° + 90°) = 25°

m∠2 = 25°.

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