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.

1. if m∠mkl = 83°, m∠jkl = 127°, and m∠jkm=(3x - 10)°, find the value of x.
2. if m∠rst=(12x - 1)°, m∠rsu=(9x - 15)°, and m∠ust = 53°, find each measure.

Explanation:

Step1: Recall angle - vertex definition

The vertex of an angle is the common endpoint of the rays that form the angle. For ∠1, the vertex is B.

Step2: Identify angle - sides

The sides of ∠1 are rays BA and BC.

Step3: Find alternative angle name

Another name for ∠5 can be ∠EBD since the vertex is B and the rays are BE and BD.

Step4: Classify angles

• ∠FBC is a right - angle (since BF⊥AC).
• ∠EBF is an obtuse angle (greater than 90°).
• ∠ABC is a straight - angle (180° as BF⊥AC and ∠ABF + ∠FBC=180°).

Step5: Determine angle bisector

If there is no clear indication from the diagram, we assume no angle bisector for now. But if we consider symmetry or equal - angle markings in a proper diagram, we would identify it. Here, no information is given, so we can't name one.

Step6: Calculate angle measure

If m∠EBD = 36° and m∠DBC = 108°, then m∠EBC=m∠EBD + m∠DBC=36°+108° = 144°.

Step7: Calculate another angle measure

If m∠EBF = 117° and ∠EBF+∠ABE = 180° (linear pair), then m∠ABE=180° - 117° = 63°.

For the second part:
If m∠MKL = 83°, m∠JKL = 127°, and m∠JKM=(3x - 10)°, then m∠JKL=m∠JKM + m∠MKL.
So, 127°=(3x - 10)°+83°.

Step1: Simplify the equation

127=3x - 10+83.
127=3x + 73.

Step2: Solve for x

Subtract 73 from both sides: 127 - 73=3x, 54 = 3x.
Divide both sides by 3: x = 18.

Answer:

a) B
b) Rays BA and BC
c) ∠EBD
d) ∠FBC: right - angle; ∠EBF: obtuse angle; ∠ABC: straight - angle
e) No angle bisector identified
f) 144°
g) 63°
For the second part: x = 18