Name: Date: Unit 10: Circles Homework 2: Central Angles & Arc Measures Per: ** This is a 2-page document!
Directions: Find the following are measures.
127
164
mJL = mBC mABC BC
D TO MOR C U 104
44 DE mFE- DEF- MCFD = mDFE G E
MSOR mROT – 5.
108
MYU = 67
Y XW 55 M LON MOM- MKNZ mNL – Z o MXVW VW = mYWU = N
Directions: Find the value of x.
(16x-13 31 P S T 83″
(9x + 23) @ R X
Given: CH bisects ZDHG. MGF – (2x + 6). and mDC – (x-1 (212-9 D c X G Gre Whion gebuc).
The Correct Answer and Explanation is:
.
Part 1: Find the following arc or angle measures
The text seems a bit jumbled, so I’ll correct and organize it into readable problems based on typical circle geometry questions. Then I’ll solve them and explain the concepts.
Problem Set 1 – Arc and Central Angle Measures:
- m∠JL=mBC⌢m\angle J L = m\overset{\frown}{BC}, m∠ABC=BC⌢m\angle ABC = \overset{\frown}{BC}
- m∠D=104∘m\angle D = 104^\circ
- m∠DE=44∘m\angle DE = 44^\circ, find mFE⌢m\overset{\frown}{FE}, m∠DEFm\angle DEF, mCFD⌢m\overset{\frown}{CFD}, and m∠DFEm\angle DFE
- m\angle SOR = \_\_\_), \( m\angle ROT = 5^\circ, m∠YU=67∘m\angle YU = 67^\circ, m∠XW=55∘m\angle XW = 55^\circ
- ( m\angle LON = ___), ( m\angle MOM = ___), ( m\angle KNZ = ___), ( m\angle NLZ = ___), ( m\angle XVW = ___), ( m\angle VWU = ___), ( m\angle YWU = ___)
You might want to upload or describe the associated diagram so I can solve each item with precision. Circle geometry questions like these depend heavily on the figure. That said, I’ll explain the typical way to approach them.
Concepts for Central Angles & Arc Measures
- Central Angle: An angle whose vertex is the center of a circle and whose sides are radii.
- Arc Measure: Equal to the measure of the central angle that intercepts the arc.
- If a central angle is θ∘\theta^\circ, the arc it intercepts is also θ∘\theta^\circ.
- The sum of all central angles in a circle is 360°.
- If you’re given expressions like (16x−13)∘(16x – 13)^\circ and another angle 83∘83^\circ, and they’re part of a semicircle or a full circle, set up the equation accordingly.
Part 2: Solve for x
Problem:
16x−13=∠PST16x – 13 = \angle PST
83∘83^\circ is the measure of the remaining angle
Probably, both are parts of a circle totaling 180∘180^\circ or 360∘360^\circ
Let’s assume these angles are central angles that sum to 180°: (16x−13)+83=18016x+70=18016x=110x=11016=6.875(16x – 13) + 83 = 180 \\ 16x + 70 = 180 \\ 16x = 110 \\ x = \frac{110}{16} = 6.875
Part 3: Geometry Proof/Analysis
Given: CH bisects ∠DHG, m∠MGF = (2x + 6), and m∠DC = (x – 1)
Assume you’re asked to find x or prove congruent angles based on a bisected angle:
If CH bisects ∠DHG, then: ∠DHC=∠CHG\angle DHC = \angle CHG
If angle measures are expressed as algebraic expressions, set them equal to each other and solve for x.
Let’s assume ∠DHC=2x+6\angle DHC = 2x + 6, and ∠CHG=x−1\angle CHG = x – 1 2x+6=x−12x−x=−1−6x=−72x + 6 = x – 1 \\ 2x – x = -1 – 6 \\ x = -7
Plug in to verify: ∠DHC=2(−7)+6=−14+6=−8∘(which is invalid)\angle DHC = 2(-7) + 6 = -14 + 6 = -8^\circ \quad \text{(which is invalid)}
So it’s likely something’s off here—either the problem is miswritten or the expressions belong to different angles.
Explanation (Sample)
Understanding central angles and arc measures is essential in circle geometry. A central angle has its vertex at the center of a circle and its sides are radii, which directly determines the arc it intercepts. The arc measure is equal to the central angle. For example, if a central angle measures 60°, the arc it intercepts also measures 60°. This is a foundational concept used in many geometry problems involving circles.
In problems involving algebraic expressions like 16x−1316x – 13 and 83∘83^\circ, we often set up equations that total 180° or 360°, depending on whether they form a semicircle or full circle. By solving these equations, we find the value of xx, which then allows us to compute unknown angle or arc measures.
In more advanced problems, like those involving angle bisectors, understanding that the bisector divides the angle into two equal parts is crucial. If an angle is bisected, the two resulting angles are congruent. For instance, if a bisector divides angle ∠DHG and one part is expressed as 2x+62x + 6 while the other is x−1x – 1, setting these equal and solving gives the correct value of xx.
Circle geometry blends algebra and visual reasoning. With clear diagrams and proper setup, even complex-looking problems can be simplified into solvable equations. It’s always essential to verify answers by substituting back and ensuring the measures make logical sense (i.e., angles should not be negative).
