Question

1. find the values of x and y in the diagram below. solve for x first

Explanation:

Step1: Solve for \( x \) using vertical angles and triangle angle - sum? Wait, first, the vertical angles: the angle \( (6x - 11)^\circ \) and the angle opposite? Wait, no, let's look at the triangles. Wait, actually, the two triangles are congruent? Wait, no, let's use the fact that vertical angles are equal? Wait, no, maybe the triangles are similar or we can use the fact that in a triangle, the sum of angles is \( 180^\circ \). Wait, first, let's find \( x \). Let's look at the angles involving \( x \). The angle \( (6x - 11)^\circ \) and the angle in the other triangle: wait, maybe the two triangles are congruent, so the corresponding angles are equal. Wait, the angle \( (x + 5)^\circ \) and \( (y + 14)^\circ \)? No, wait, let's start with \( x \). Let's assume that the two triangles are congruent, so the angle \( (6x - 11)^\circ \) and the angle adjacent to \( 81^\circ \): wait, no, let's use the fact that in a triangle, the sum of angles is \( 180^\circ \). Wait, first, solve for \( x \). Let's consider the triangle with angle \( 81^\circ \), \( (x + 5)^\circ \), and the vertical angle of \( (6x - 11)^\circ \). Wait, vertical angles are equal, so the angle opposite to \( (6x - 11)^\circ \) is equal to it. Then, in the triangle with angles \( 81^\circ \), \( (x + 5)^\circ \), and \( (6x - 11)^\circ \), the sum of angles is \( 180^\circ \). So:

\( 81+(x + 5)+(6x - 11)=180 \)

Simplify the left - hand side:

\( 81+x + 5+6x-11 = 180 \)

Combine like terms:

\( (x+6x)+(81 + 5-11)=180 \)

\( 7x+75 = 180 \)

Subtract 75 from both sides:

\( 7x=180 - 75=105 \)

Divide both sides by 7:

\( x=\frac{105}{7}=15 \)

Step2: Now that we have \( x = 15 \), we can find the angle \( (x + 5)^\circ=(15 + 5)^\circ = 20^\circ \), and \( (6x - 11)^\circ=(6\times15-11)^\circ=(90 - 11)^\circ = 79^\circ \)

Now, let's look at the other triangle with angles \( (4y-18)^\circ \), \( (y + 14)^\circ \), and \( (6x - 11)^\circ=79^\circ \). The sum of angles in a triangle is \( 180^\circ \), so:

\( (4y-18)+(y + 14)+79 = 180 \)

Combine like terms:

\( 4y-18+y + 14+79 = 180 \)

\( 5y+( - 18+14 + 79)=180 \)

\( 5y+( - 4+79)=180 \)

\( 5y + 75=180 \)

Subtract 75 from both sides:

\( 5y=180 - 75 = 105 \)

Divide both sides by 5:

\( y=\frac{105}{5}=21 \)

Wait, let's check again. Wait, maybe the two triangles are congruent, so the angles \( (4y - 18)^\circ=81^\circ \)? Wait, that might be a better approach. Let's try that. If \( (4y-18)^\circ = 81^\circ \), then:

\( 4y-18 = 81 \)

\( 4y=81 + 18=99 \)

\( y=\frac{99}{4}=24.75 \), which doesn't match the previous result. So my first approach was wrong.

Let's start over. The two triangles are congruent (since they are formed by vertical angles, so the included angles are equal, and if the sides are equal, but maybe the triangles are congruent by ASA). So the angle \( (4y - 18)^\circ \) corresponds to \( 81^\circ \), the angle \( (y + 14)^\circ \) corresponds to \( (x + 5)^\circ \), and the angle \( (6x - 11)^\circ \) corresponds to the angle opposite (vertical angle). Wait, vertical angles are equal, so \( (6x - 11)^\circ \) is equal to the angle in the other triangle. Let's use the fact that in each triangle, the sum of angles is \( 180^\circ \).

For the right - hand triangle: angles are \( 81^\circ \), \( (x + 5)^\circ \), and \( \theta \) (where \( \theta=(6x - 11)^\circ \) because vertical angles are equal). So \( 81+(x + 5)+(6x - 11)=180 \)

Simplify:

\( 81+x + 5+6x-11 = 180 \)

\( 7x+75 = 180 \)

\( 7x=105 \)

\( x = 15 \) (this part was correct)

Now, for the left - hand triangle: angles are \( (4y - 18)^\circ \), \( (y + 14)^\circ \), and \( (6x - 11)^\circ=(6\times15 - 11)^\circ=79^\circ \)

So \( (4y-18)+(y + 14)+79 = 180 \)

\( 5y+75 = 180 \)

\( 5y=105 \)

\( y = 21 \)

Wait, but let's check with the corresponding angles. If \( x = 15 \), then \( (x + 5)^\circ=20^\circ \), and \( (y + 14)^\circ=(21 + 14)^\circ=35^\circ \), which are not equal. So my assumption of congruence is wrong.

Wait, maybe the triangles are similar. Let's use the fact that vertical angles are equal, so the included angles are equal, so the triangles are similar by AA similarity. So the corresponding angles are equal.

So \( (4y - 18)^\circ=81^\circ \) (corresponding angles)

\( 4y=81 + 18=99 \)

\( y=\frac{99}{4}=24.75 \)

\( (y + 14)^\circ=(24.75+14)^\circ=38.75^\circ \)

\( (x + 5)^\circ=(x + 5)^\circ \), and \( (x + 5)^\circ \) should correspond to \( (y + 14)^\circ \)? No, that's not right.

Wait, let's look at the diagram again. The two triangles have a pair of vertical angles (the angle \( (6x - 11)^\circ \) and its vertical opposite). So in triangle 1: angles are \( (4y - 18)^\circ \), \( (y + 14)^\circ \), and \( (6x - 11)^\circ \)

In triangle 2: angles are \( 81^\circ \), \( (x + 5)^\circ \), and \( (6x - 11)^\circ \) (since vertical angles are equal)

So for triangle 2: \( 81+(x + 5)+(6x - 11)=180 \)

As before:

\( 81+x + 5+6x-11 = 180 \)

\( 7x+75 = 180 \)

\( 7x=105 \)

\( x = 15 \)

Now, since the two triangles are congruent (because they have two angles equal: the vertical angle and another angle? Wait, no, if two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. But if the sides are equal, they are congruent. But we can also use the fact that in triangle 1, \( (4y - 18)+(y + 14)+(6x - 11)=180 \), and in triangle 2, \( 81+(x + 5)+(6x - 11)=180 \). So we can set the sum of the first two angles in each triangle equal:

\( (4y - 18)+(y + 14)=81+(x + 5) \)

We know \( x = 15 \), so:

\( 5y-4=81 + 20 \)

\( 5y-4=101 \)

\( 5y=105 \)

\( y = 21 \)

Now, check the angles:

For \( x = 15 \):

\( x + 5=20^\circ \), \( 6x - 11=79^\circ \)

For \( y = 21 \):

\( y + 14=35^\circ \), \( 4y - 18=4\times21-18 = 84 - 18=66^\circ \)

Now, check triangle 1: \( 66+35+79 = 180 \) (66 + 35 = 101, 101+79 = 180)

Triangle 2: \( 81+20+79 = 180 \) (81 + 20 = 101, 101+79 = 180)

Perfect, so the calculations are correct.

Step1: Solve for \( x \)

In the right - hand triangle, the sum of interior angles is \( 180^\circ \). The angles are \( 81^\circ \), \( (x + 5)^\circ \), and \( (6x - 11)^\circ \) (since vertical angles are equal). So we set up the equation:

\( 81+(x + 5)+(6x - 11)=180 \)

Simplify the left - hand side:

\( 81+x + 5+6x-11=7x + 75 \)

So, \( 7x+75 = 180 \)

Subtract 75 from both sides:

\( 7x=180 - 75=105 \)

Divide both sides by 7:

\( x=\frac{105}{7}=15 \)

Step2: Solve for \( y \)

In the left - hand triangle, the sum of interior angles is \( 180^\circ \). The angles are \( (4y - 18)^\circ \), \( (y + 14)^\circ \), and \( (6x - 11)^\circ \). We know \( x = 15 \), so \( 6x - 11=6\times15-11 = 79^\circ \)

Set up the equation for the sum of angles in the left - hand triangle:

\( (4y - 18)+(y + 14)+79 = 180 \)

Simplify the left - hand side:

\( 4y-18+y + 14+79=5y + 75 \)

So, \( 5y+75 = 180 \)

Subtract 75 from both sides:

\( 5y=180 - 75=105 \)

Divide both sides by 5:

\( y=\frac{105}{5}=21 \)

Answer:

\( x = 15 \), \( y = 21 \)