Question

1. in this figure, ( mangle3 = x + 18 ), ( mangle4 = x + 15 ), and ( mangle5 = 4x + 3 ). show all work for parts a, b, and c.

a. what is the value of ( x )?
b. what is the measure of ( angle1 )?
c. is ( angle3 ) complementary to ( angle1 )? explain.

1. ( overrightarrow{mp} ) is the angle bisector of ( angle lmn ). given that ( angle lmp ) is a right angle, what type of angle is ( angle lmn )? explain how you know.
2. state the angle addition postulate in your own words.
3. a protractor is properly aligned with the vertex of ( angle klm ). ( overrightarrow{lk} ) passes through the mark for ( 38^{circ} ) on the protractor and ( overrightarrow{lm} ) passes through the mark for ( 126^{circ} ). what is ( mangle klm )?
4. ( angle gfh ) and ( angle hfj ) are adjacent and congruent. what are two conclusions you can draw from this information? support your answers.

Explanation:

Step1: Use the angle - sum property (angles around a point)

We know that \(m\angle3 + m\angle4+m\angle5 = 90^{\circ}\) (since \(\angle1\) and the sum of \(\angle3,\angle4,\angle5\) form a right - angle).
Substitute \(m\angle3=x + 18\), \(m\angle4=x + 15\), and \(m\angle5=4x+3\) into the equation:
\((x + 18)+(x + 15)+(4x+3)=90\)

Step2: Simplify the left - hand side of the equation

Combine like terms: \((x+x + 4x)+(18 + 15+3)=90\)
\(6x+36 = 90\)

Step3: Solve for \(x\)

Subtract 36 from both sides: \(6x=90 - 36\)
\(6x=54\)
Divide both sides by 6: \(x=\frac{54}{6}=9\)

Step4: Find \(m\angle1\)

First, find \(m\angle3\), \(m\angle4\), and \(m\angle5\)
\(m\angle3=x + 18=9+18 = 27^{\circ}\)
\(m\angle4=x + 15=9+15 = 24^{\circ}\)
\(m\angle5=4x+3=4\times9+3=36 + 3=39^{\circ}\)
Since \(m\angle1+ m\angle3+m\angle4+m\angle5 = 180^{\circ}\) (a straight - angle), \(m\angle1=180-(m\angle3+m\angle4+m\angle5)\)
\(m\angle1=180-(27 + 24+39)=180 - 90=90^{\circ}\)

Step5: Check if \(\angle3\) is complementary to \(\angle1\)

Two angles are complementary if their sum is \(90^{\circ}\)
\(m\angle3+m\angle1=27 + 90=117
eq90\)

Answer:

a. \(x = 9\)
b. \(m\angle1=90^{\circ}\)
c. No, because \(m\angle3+m\angle1=117^{\circ}
eq90^{\circ}\)