Question

isosceles triangles: draw a picture for each figure. then solve for x. 9. in δabc, ab = bc. if m∠b = 34°, then m∠a = __. 10. in δdef, de = df. if m∠e = 46°, then m∠d = . 11. in δghj, ∠g ≅ ∠j. if gh = 31 and the perimeter of ghj is 100, then gj = __.

Explanation:

Problem 9

Step1: Recall Isosceles Triangle Angles

In an isosceles triangle, the angles opposite equal sides are equal. Here, \( AB = BC \), so the angles opposite them (\( \angle C \) and \( \angle A \)) are equal. Let \( m\angle A = m\angle C = x \).

Step2: Use Triangle Angle Sum

The sum of angles in a triangle is \( 180^\circ \). So, \( m\angle A + m\angle B + m\angle C = 180^\circ \). Substituting values: \( x + 34^\circ + x = 180^\circ \).

Step3: Solve for x

Combine like terms: \( 2x + 34^\circ = 180^\circ \). Subtract \( 34^\circ \): \( 2x = 180^\circ - 34^\circ = 146^\circ \). Divide by 2: \( x = \frac{146^\circ}{2} = 73^\circ \).

Step1: Identify Equal Angles

In \( \triangle DEF \), \( DE = DF \), so the angles opposite them (\( \angle F \) and \( \angle E \)) are equal. Thus, \( m\angle F = m\angle E = 46^\circ \).

Step2: Apply Angle Sum Property

Sum of angles in a triangle is \( 180^\circ \). Let \( m\angle D = y \). Then \( y + 46^\circ + 46^\circ = 180^\circ \).

Step3: Solve for y

Combine terms: \( y + 92^\circ = 180^\circ \). Subtract \( 92^\circ \): \( y = 180^\circ - 92^\circ = 88^\circ \).

Step1: Identify Equal Sides

In \( \triangle GHJ \), \( \angle G \cong \angle J \), so the sides opposite them (\( HJ \) and \( GH \)) are equal. Thus, \( HJ = GH = 31 \).

Step2: Use Perimeter Formula

Perimeter \( P = GH + HJ + GJ \). Given \( P = 100 \), \( GH = 31 \), \( HJ = 31 \). Substitute: \( 100 = 31 + 31 + GJ \).

Step3: Solve for GJ

Calculate \( 31 + 31 = 62 \). Then \( GJ = 100 - 62 = 38 \).

Answer:

\( 73^\circ \)

Problem 10