Question

problem 1: two angles form a linear pair. if one angle measures 65 degrees, find the measure of the other angle. a / b linear pair problem 2 angles ∠abc and ∠cbd form a linear pair. if ∠abc=(3x + 10)° and ∠cbd=(5x - 2)°, find the value of x and the measures of both angles. problem 3 two adjacent angles form a linear pair. the measure of one angle is three times the measure of the other. find the measure of each angle.

Explanation:

Problem 1

Step1: Recall linear - pair property

The sum of angles in a linear pair is 180°.
Let the known angle be \(a = 65^{\circ}\) and the unknown angle be \(b\). Then \(a + b=180^{\circ}\).

Step2: Solve for the unknown angle

\(b = 180^{\circ}-a\). Substitute \(a = 65^{\circ}\) into the equation: \(b=180 - 65=115^{\circ}\).

Step1: Use linear - pair property

Since \(\angle ABC\) and \(\angle CBD\) form a linear pair, \(\angle ABC+\angle CBD = 180^{\circ}\).
Substitute \(\angle ABC=(3x + 10)^{\circ}\) and \(\angle CBD=(5x - 2)^{\circ}\) into the equation: \((3x + 10)+(5x - 2)=180\).

Step2: Simplify the left - hand side

Combine like terms: \(3x+5x+10 - 2=180\), which gives \(8x + 8 = 180\).

Step3: Solve for \(x\)

Subtract 8 from both sides: \(8x=180 - 8=172\). Then divide both sides by 8: \(x=\frac{172}{8}=21.5\).

Step4: Find the measure of \(\angle ABC\)

Substitute \(x = 21.5\) into \(\angle ABC=(3x + 10)^{\circ}\): \(\angle ABC=3\times21.5+10=64.5 + 10=74.5^{\circ}\).

Step5: Find the measure of \(\angle CBD\)

Substitute \(x = 21.5\) into \(\angle CBD=(5x - 2)^{\circ}\): \(\angle CBD=5\times21.5-2=107.5 - 2=105.5^{\circ}\).

Step1: Let the angles

Let one angle be \(x\) and the other be \(3x\) since one angle is three times the other and they form a linear pair.

Step2: Use linear - pair property

\(x + 3x=180^{\circ}\) (because the sum of angles in a linear pair is \(180^{\circ}\)).

Step3: Solve for \(x\)

Combine like terms: \(4x=180\), then \(x=\frac{180}{4}=45^{\circ}\).

Step4: Find the other angle

The other angle is \(3x\), so \(3x=3\times45 = 135^{\circ}\).

Answer:

\(115^{\circ}\)

Problem 2