Question

solve for x. then find the measure of the angles.

x =

m∠cab =

m∠abc =

m∠dcb =

blank 1:

blank 2:

blank 3:

blank 4:

Explanation:

Step1: Apply the exterior angle theorem

The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. So, \(18x - 15=(13x - 11)+(4x + 1)\).

Step2: Simplify the right - hand side of the equation

Simplify \((13x - 11)+(4x + 1)\):

$$
LATEXBLOCK0
$$

The equation becomes \(18x - 15=17x-10\).

Step3: Solve for \(x\)

Subtract \(17x\) from both sides: \(18x-17x-15=17x - 17x-10\), which gives \(x-15=-10\).
Add \(15\) to both sides: \(x=-10 + 15\), so \(x = 5\).

Step4: Find \(m\angle CAB\)

Substitute \(x = 5\) into \(13x-11\). Then \(m\angle CAB=13x-11=13\times5-11=65 - 11=54^{\circ}\).

Step5: Find \(m\angle ABC\)

Substitute \(x = 5\) into \(4x + 1\). Then \(m\angle ABC=4x + 1=4\times5+1=20 + 1=21^{\circ}\).

Step6: Find \(m\angle DCB\)

Substitute \(x = 5\) into \(18x-15\). Then \(m\angle DCB=18x-15=18\times5-15=90-15=75^{\circ}\).

Answer:

Blank 1: \(5\)
Blank 2: \(54^{\circ}\)
Blank 3: \(21^{\circ}\)
Blank 4: \(75^{\circ}\)