Question

1. use the diagram below to complete each part.

a) name the vertex of ∠4.
b) name the sides of ∠1.
c) write another name for ∠5.
d) classify each angle: ∠fbc: ∠ebf: ∠abc:
e) name an angle bisector.
f) if m∠ebd = 36° and m∠dbc = 108°, find m∠ebc.
g) if m∠ebf = 117°, find m∠abe.

• (overline{bf}perpoverline{ac})

1. if m∠mkl = 83°, m∠jkl = 127°, and m∠jkm=(9x - 10)°, find the value of x.
2. if m∠rst=(12x - 1)°, m∠rsu=(9x - 15)°, and m∠ust = 53°, find each measure.

x =
m∠rst =
m∠rsu =

Explanation:

Step1: Identify vertex for ∠4

The vertex of an angle is the common endpoint. For ∠4, the vertex is B.

Step2: Name sides of ∠1

The sides of an angle are the two rays that form the angle. For ∠1, the sides are $\overrightarrow{BA}$ and $\overrightarrow{BC}$.

Step3: Another name for ∠5

∠5 can also be named as ∠ABE.

Step4: Classify angles

• ∠FBC: Acute (since it's less than 90° as $\overrightarrow{BF}\perp\overrightarrow{AC}$ and ∠FBC is part of the right - angle formed by $\overrightarrow{BF}$ and $\overrightarrow{AC}$).
• ∠EBF: Obtuse (since it's greater than 90°).
• ∠ABC: Straight (it appears to be a 180 - degree angle as it's a flat angle).

Step5: Name angle bisector

Since $\overrightarrow{BF}\perp\overrightarrow{AC}$, if the figure is symmetric about $\overrightarrow{BF}$, $\overrightarrow{BF}$ could be an angle bisector of ∠ABC (assuming equal angles on either side of $\overrightarrow{BF}$ in the context of the right - angle at B with respect to $\overrightarrow{AC}$).

Step6: Find m∠EBC

We know that ∠EBC=∠EBD + ∠DBC. Given m∠EBD = 36° and m∠DBC = 108°, then m∠EBC=36°+ 108° = 144°.

Step7: Find m∠ABE

We know that ∠EBF=∠ABE + ∠ABF. Since $\overrightarrow{BF}\perp\overrightarrow{AC}$, ∠ABF = 90°. Given m∠EBF = 117°, then m∠ABE=m∠EBF - 90°=117° - 90° = 27°.

Step8: Solve for x in ∠JKM problem

We know that ∠JKL=∠JKM+∠MKL. Given m∠MKL = 83°, m∠JKL = 127°, and m∠JKM=(9x - 10)°. So, (9x - 10)+83 = 127.
First simplify the left - hand side: 9x+73 = 127.
Then subtract 73 from both sides: 9x=127 - 73 = 54.
Divide both sides by 9: x = 6.

Step9: Solve for x in ∠RST problem

We know that ∠RST=∠RSU+∠UST. Given m∠RST=(12x - 1)°, m∠RSU=(9x - 15)°, and m∠UST = 53°. So, (12x - 1)=(9x - 15)+53.
First simplify the right - hand side: (12x - 1)=9x+38.
Subtract 9x from both sides: 12x-9x - 1=38.
3x-1 = 38.
Add 1 to both sides: 3x=39.
Divide both sides by 3: x = 13.
Then find m∠RST: m∠RST=12x - 1=12×13 - 1=156 - 1 = 155°.
Find m∠RSU: m∠RSU=9x - 15=9×13 - 15=117 - 15 = 102°.

Answer:

a) B
b) $\overrightarrow{BA}$, $\overrightarrow{BC}$
c) ∠ABE
d) ∠FBC: Acute, ∠EBF: Obtuse, ∠ABC: Straight
e) $\overrightarrow{BF}$
f) 144°
g) 27°

1. x = 6
2. x = 13, m∠RST = 155°, m∠RSU = 102°