Question

independent practice: isosceles & equilateral triangles
12 ∠2 = ______ find the measure of ∠2. m∠2 = 4x - 2
13 x = ______ find the measure of x.

Explanation:

12. Find the measure of \( \angle 2 \)

Step1: Find the base - angle of the isosceles triangle

In an isosceles triangle, the base - angles are equal. Let's first consider the left - hand isosceles triangle. The sum of the interior angles of a triangle is \( 180^{\circ} \). If one of the base - angles is \( 68^{\circ} \), then the vertex - angle of this isosceles triangle is \( 180^{\circ}-2\times68^{\circ}=180^{\circ} - 136^{\circ}=44^{\circ} \).

Step2: Use the property of the equilateral triangle

The right - hand triangle is equilateral (all sides are equal, so all angles are \( 60^{\circ} \)). But we are interested in the angle \( \angle 2 \). The angle adjacent to \( \angle 2 \) in the combined figure (from the equilateral triangle part) is \( 60^{\circ} \).
Since the vertex - angle of the left - hand isosceles triangle is \( 44^{\circ} \), and we know that \( m\angle2 = 4x - 2 \). But we can also find \( \angle 2 \) directly. The angle adjacent to \( \angle 2 \) (from the non - equilateral part) is \( 44^{\circ} \), and from the equilateral part (angles of an equilateral triangle are \( 60^{\circ} \)), we use the fact that the sum of angles around a point (at the vertex of the combined figure) is not relevant here. Wait, no, re - considering:
The left - hand isosceles triangle: vertex - angle \( v=180 - 2\times68=44^{\circ} \). The right - hand triangle (equilateral) has all angles \( 60^{\circ} \). But in the combined figure, for the angle at the top:
\( m\angle2=60 - 44=16^{\circ} \).
Alternatively, if we assume the left - hand isosceles triangle (two sides equal) and then using the angle sum property. Wait, another approach:
The left - hand isosceles triangle: base - angles \( 68^{\circ} \) each. Let the vertex - angle of the left - hand isosceles triangle be \( A \). Then \( A=180-(68 + 68)=44^{\circ} \).
The right - hand triangle (equilateral, so each angle \( 60^{\circ} \)). The angle \( \angle 2 \) is \( 60 - 44=16^{\circ} \).
If we use the formula \( m\angle2 = 4x-2 \), but we can also check:
Let's assume the left - hand isosceles triangle (two equal sides). The sum of angles in a triangle is \( 180^{\circ} \). Then the vertex - angle of the left - hand isosceles triangle \(=180 - 2\times68 = 44^{\circ}\).
The right - hand triangle (equilateral, angle \( 60^{\circ} \)). So \( m\angle2=60 - 44=16^{\circ}\).

13. Find the value of \( x \)

Step1: Use the property of vertical angles and right - angled triangles

In the lower triangle (right - angled triangle with one angle \( 60^{\circ} \)), the third angle (let's call it \( y \)) is \( 180-(90 + 60)=30^{\circ}\).
Since the two triangles are related by vertical angles. The angle \( x \) and the sum of angles (using the property of vertical angles and angle sum of a triangle).
Another approach:
We know that in a triangle, the sum of interior angles is \( 180^{\circ} \).
In the lower right - angled triangle, one angle is \( 90^{\circ} \) and another is \( 60^{\circ} \). The third angle (let's call it \( z\)) is \( 180-(90 + 60)=30^{\circ}\).
Using the property of vertical angles (opposite angles formed by two intersecting lines are equal).
The upper triangle: two sides are equal (marked). Let's use the angle sum property.
The angle adjacent to \( x \) (using vertical angles) from the lower triangle is \( 30^{\circ} \).
The sum of angles in the upper triangle: \( 180^{\circ} \). If two sides are equal (isosceles triangle), and using the vertical angles.
\( x=180-(30 + 30)=120^{\circ}\)

Answer:

1. \( 16^{\circ}\)
2. \( 120^{\circ}\)