Question

1. find the measures of the angles of a right triangle whose acute angles have measures of 5x and 4x.

a. 40, 50, 90
b. 25, 40, 65
c. 30, 45, 60
d. 45, 60, 90
e. none of the above

1. find the measures of the angles of a triangle whose angles have measures of x - 20, 3x, and 4x.

a. 35, 55, 90
b. 5, 40, 65
c. 15, 25, 45
d. 5, 75, 100
e. none of the above

1. a polygon is a geometric figure whose sides are ______.

a. lines
b. line segments
c. rays
d. equilateral
e. congruent

1. in a(n) ______ triangle, every angle is less than 90 degrees.

a. scalene
b. right
c. isosceles
d. obtuse
e. acute

1. the measure of each angle of an equiangular triangle is ______.

a. 80
b. 180
c. different
d. 60
e. 90

1. ____ of a polygon is an angle that forms a linear pair with one of the ____ of the polygon.

a. an interior angle; exterior angles
b. a remote angle; exterior angles
c. an exterior angle; interior angles
d. the largest angle; acute angles
e. an exterior angle; remote interior angles

Explanation:

6.

Step1: Use the angle - sum property of a right - triangle

In a right - triangle, the sum of the two acute angles is \(90^{\circ}\). Given the acute angles are \(5x\) and \(4x\). So, \(5x + 4x=90\).

Step2: Solve the equation for \(x\)

Combine like terms: \(9x = 90\). Then, divide both sides by \(9\): \(x=\frac{90}{9}=10\).

Step3: Find the measures of the acute angles

Substitute \(x = 10\) into \(5x\) and \(4x\). For \(5x\), we have \(5\times10 = 50\). For \(4x\), we have \(4\times10=40\). And the right - angle is \(90^{\circ}\).

A polygon is defined as a geometric figure whose sides are line segments. A line has infinite length, a ray has one - end infinite, equilateral and congruent refer to side - length properties (equilateral for equal side lengths in a special case, congruent for equal in general, but not the basic definition of a side of a polygon).

An acute triangle is defined as a triangle where every angle is less than \(90^{\circ}\). A scalene triangle is defined by side lengths (all sides are different), a right triangle has one \(90^{\circ}\) angle, an isosceles triangle has at least two equal sides, and an obtuse triangle has one angle greater than \(90^{\circ}\).

The sum of the interior angles of a triangle is \(180^{\circ}\). In an equiangular triangle, all angles are equal. Let each angle be \(x\). Then \(3x=180\), so \(x = 60^{\circ}\).

Answer:

A. \(40,50,90\)

7.

Step1: Use the angle - sum property of a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \((x - 20)+3x + 4x=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(x-20 + 3x+4x=(1 + 3+4)x-20=8x-20\). Then the equation becomes \(8x-20 = 180\).

Step3: Solve for \(x\)

Add \(20\) to both sides: \(8x=180 + 20=200\). Divide both sides by \(8\): \(x=\frac{200}{8}=25\).

Step4: Find the measures of the angles

For \(x-20\): \(25-20 = 5\). For \(3x\): \(3\times25 = 75\). For \(4x\): \(4\times25=100\).