10th Grade Math Ch. 6,7, and 8

Angles 1 and 2 are both complementary to ∠3. What is m∠1 if m∠2 = (2x)° and m∠3 = (-x + 70)°?

• 50 degrees

• 40 degrees

• 30 degrees

• 90 degrees

Answer explanation

$\angle2$ $\angle1$ is complementary to $\angle3$

$\angle2$ is complementary to $\angle3$
Therefore $\angle1$ = $\angle2$
$\angle2$ + $\angle3$ = $90\degree$
2x + (-x +70) = 90
2x - x + 70 = 90
x + 70 = 90
x = 90 - 70
x = 20
$\angle2$ = 2x
2(20) = $\angle2$
40 = $\angle1$

In triangle ABC, the point I is the midpoint of segment BC, and J is the centroid. Find x if AJ = 3x-3 and AI = 6 + x.

• 6

• 4

• 8

• 3

Answer explanation

The centroid of a triangle theorem states that

$AJ=\frac{2}{3}AI$
$3x-3=\frac{2}{3}\left(6+x\right)$

$3x-3=\frac{12}{3}+\frac{2}{3}x$
$3\left(3x-3=\frac{12}{3}+\frac{2}{3}x\right)3$
$9x-9=12+2x$
$\frac{7x}{7}=\frac{21}{7}$
x=3

Find x.

• 5

• 6

• 4

• 3

Answer explanation

Theorem 7.6: If a line passes through the midpoint of one side of a triangle and is parallel to another side, then it cuts the third side of the triangle at its midpoint.
So 9x-24 is congruent to 3x
9x-24=3x
9x-3x=24
6x=24
x= 24/6
x=4

Solve for x and y.

• x=9, y=80

• x=12, y=90

• x=7, y=100

• x=5, y=85

Answer explanation

Solve for the vertical angles.

4x-6 = 2x +12

4x-2x=12+6

2x=18

x=18/2

x=9

Replace the x for 9 in (5x+35)

Solve for vertical angles

5(9) + 35 = y

45 + 35 = y

80 = y

The two lines are parallel.

Solve for x.

• 60

• 146

• 36

• 70

Answer explanation

Through the angle, x, draw a line.

This can be solved using the fact that transversal and parallel lines form interior angles on the same side of the transversal that are supplementary.

Where the line is drawn, x is now split.

Supplementary angle rules help us to solve for the two angles

Then add those two to solve for x.

The equation in a is 2a = _________ - ___________

• 180 -10

• 150 - a

• 160 - a

• 150 - 22

Answer explanation

We can find that these two angles equal each other through alternate interior angles and corresponding angles.

The value of a is_________________

• 10

• 20

• 40

• 50

Answer explanation

2a=150-a

3a=150

a=150/3

a=50

Find the measure of angles 1 and 4.

• angle 1 = 60, angle 4 = 30

• angle 1 = 50, angle 4 = 20

• angle 1 = 70, angle 4 = 40

• angle 1 = 80, angle 4 = 10

Answer explanation

$\angle1\cong\angle2$ (base angles of an isoceles triangle are congruent)
$\angle1+x=2x$ (exterior angle theorem)
$\angle1=x$
since angle 1 is congruent to angle 2
$\angle2+2x=180$ (angles in a linear pair are supplementary)
x+2x=180
3x=180
x=180/3
x=60
$2x+\angle3+\angle4=180$ (angle sum theorem)
$2\left(60\right)+\angle3+\angle4=180$
$\angle3+\angle4=180-60$
$\angle3\cong\angle4$
Since angle 3 and 4 are congruent we can divide 120 by 2 each angle equals 30

solve for x.

• 10

• 15

• 20

• 25

Answer explanation

angle sum theorem

(x+40) + (x+20)+90=180

2x+150=180

2x=30

x=15

B is equidistant from what two points.

• A and O

• C and O

• A and C

Answer explanation

Equidistant means the same distant.