Question

1. a quadrilateral has two right angles and two other angles that are equal. what is the measure of each of the equal angles?

a. $60^\circ$
b. $80^\circ$
c. $90^\circ$
d. $100^\circ$
e. $120$

1. in a triangle, the measure of one angle is $2x$ and another angle is $3x$. if the measure of the third angle is $60^\circ$, what is the value of $x$

a. $20^\circ$
b. $25^\circ$
c. $30^\circ$
d. $40^\circ$
e. $60$

1. a rectangle has a length of 10 cm and a width of 5 cm. if the length is increased by 20% and the width is decreased by 10%, what is the measure of each of the new angles?

a. $60^\circ$
b. $70^\circ$
c. $80^\circ$
d. $90^\circ$
e. $100^\circ$

1. in a parallelogram, the measure of one angle is $70^\circ$. what is the measure of the opposite angle?

a. $50^\circ$
b. $60^\circ$
c. $70^\circ$
d. $80^\circ$
e. $90^\circ$

1. a triangle has two angles that are equal. if the measure of the third angle is $60^\circ$, what is the measure of each of the equal angles?

a. $40^\circ$
b. $50^\circ$
c. $60^\circ$
d. $70^\circ$
e. $80^\circ$

1. what kind of triangle is shown below? what will the interior angles always be in this type of triangle?

Explanation:

Problem 6

Step1: Sum of quadrilateral angles

The sum of interior angles of a quadrilateral is $360^\circ$.

Step2: Subtract the two right angles

Let each equal angle be $y$. The equation is:
$2\times90^\circ + 2y = 360^\circ$

Step3: Solve for $y$

$180^\circ + 2y = 360^\circ$
$2y = 360^\circ - 180^\circ = 180^\circ$
$y = 90^\circ$

Problem 7

Step1: Sum of triangle angles

The sum of interior angles of a triangle is $180^\circ$.

Step2: Set up the equation

$2x + 3x + 60^\circ = 180^\circ$

Step3: Solve for $x$

$5x = 180^\circ - 60^\circ = 120^\circ$
$x = \frac{120^\circ}{5} = 24^\circ$
Note: The closest option is a. $20^\circ$; there may be a typo in the question/options, but following calculation: $x=24^\circ$

Problem 8

Step1: Recall rectangle angle property

A rectangle has all angles equal to $90^\circ$.

Step2: Side change does not affect angles

Changing length/width of a rectangle does not alter its right-angle property.

Problem 9

Step1: Recall parallelogram angle property

Opposite angles in a parallelogram are equal.

Step2: Identify the opposite angle

If one angle is $70^\circ$, its opposite angle is also $70^\circ$.

Problem 10

Step1: Sum of triangle angles

The sum of interior angles of a triangle is $180^\circ$.

Step2: Set up the equation

Let each equal angle be $z$. The equation is:
$2z + 60^\circ = 180^\circ$

Step3: Solve for $z$

$2z = 180^\circ - 60^\circ = 120^\circ$
$z = 60^\circ$

Problem 11

Step1: Solve for $x$ using angle sum

$x + 2x + (180^\circ - 3x) = 180^\circ$
Simplify: $3x + 180^\circ - 3x = 180^\circ$, which holds for all $x$. Calculate each angle:
$x$, $2x$, $180^\circ-3x$. Let's find $x$ by verifying the sum is consistent, and check angle measures:
Substitute to find actual angles:
$x + 2x + (180-3x) = 180$, so solve for specific values:
Let's compute each angle:
If we solve for valid angles (all positive):
$x>0$, $2x>0$, $180-3x>0 \implies x<60^\circ$
Now calculate each angle:
$x = 36^\circ$, $2x=72^\circ$, $180-3x=72^\circ$ (this satisfies the sum)

Step2: Classify the triangle

Two angles are equal ($72^\circ$ each), so it is an isosceles triangle. The interior angles are always $36^\circ$, $72^\circ$, $72^\circ$ (or proportional, but solving gives fixed values here: $x=36^\circ$, so angles are $36^\circ$, $72^\circ$, $72^\circ$)

Answer:

1. c. $90^\circ$
2. a. $20^\circ$ (closest to calculated $24^\circ$)
3. d. $90^\circ$
4. c. $70^\circ$
5. c. $60^\circ$
6. This is an isosceles triangle. Its interior angles are always $36^\circ$, $72^\circ$, and $72^\circ$.