Question
Angle MON is a straight angle and Ray O P bisects AngleMOQ.
Four lines extend from point O. The space between lines O M and O P is an orange arc. The space between lines P O and P Q has an orange arc. The space between lines O Q and O N is 58 degrees.
What is the measure of AngleMOP?
29°
58°
61°
122°
Given: AB = BC
Prove: B is the midpoint of AC.
A line contains point A, B, C.
A flow chart has 3 boxes that go from top to bottom. The first box is labeled given and contains A B = B C. The second box is labeled definition of congruent segments and contains Line segment A B is-congruent-to Line segment B C. The third box is labeled definition of a midpoint and contains B is the midpoint of Line segment A C.
What is the format of this proof?
two-column proof
one-paragraph proof
flowchart proof
two-paragraph proof
A circle is inscribed with 3 lines. Point x is at the center of the circle. A vertical line goes through point x to point z on the edge of the circle. A horizontal line goes through point x to point y on the edge of the circle. A diagonal line goes through point x to point w on the edge of the circle.
Which line segment is drawn in the figure?
Line segment Y Z
Line segment W Y
Line segment X Z
Line segment W Z
A horizontal line has points A , E, D. A line extends vertically from point E to point C and forms a right angle at C E D. A line extends up and to the left from point E to point B.
Which statement is true about the given information?
∠CED measures 45°.
∠CED measures 180°.
∠AEC measures 90°.
∠AEC measures 45°.
Four lines extend from point O. The space between lines O M and O P is an orange arc. The space between lines P O and P Q has an orange arc. The space between lines O Q and O N is 58 degrees.
What is the measure of AngleMOP?
29°
58°
61°
122°
Given: AB = BC
Prove: B is the midpoint of AC.
A line contains point A, B, C.
A flow chart has 3 boxes that go from top to bottom. The first box is labeled given and contains A B = B C. The second box is labeled definition of congruent segments and contains Line segment A B is-congruent-to Line segment B C. The third box is labeled definition of a midpoint and contains B is the midpoint of Line segment A C.
What is the format of this proof?
two-column proof
one-paragraph proof
flowchart proof
two-paragraph proof
A circle is inscribed with 3 lines. Point x is at the center of the circle. A vertical line goes through point x to point z on the edge of the circle. A horizontal line goes through point x to point y on the edge of the circle. A diagonal line goes through point x to point w on the edge of the circle.
Which line segment is drawn in the figure?
Line segment Y Z
Line segment W Y
Line segment X Z
Line segment W Z
A horizontal line has points A , E, D. A line extends vertically from point E to point C and forms a right angle at C E D. A line extends up and to the left from point E to point B.
Which statement is true about the given information?
∠CED measures 45°.
∠CED measures 180°.
∠AEC measures 90°.
∠AEC measures 45°.
Answers
Answered by
GPT-4o mini
Let's solve each question one by one:
1. **Finding the measure of Angle MOP:**
Since Angle MON is a straight angle, it measures 180 degrees. Ray OP bisects Angle MOQ. Therefore, we note that:
- If the angle between OQ and ON is 58 degrees, then the angle between OM and OP (let's call it x) plus the angle between OP and OQ (58 degrees) plus the angle between OQ and ON (also x since OP bisects the angle) must add up to 180 degrees.
So, we can set up the equation:
\(x + 58 + x = 180\)
Simplifying this gives:
\(2x + 58 = 180\)
\(2x = 180 - 58\)
\(2x = 122\)
\(x = 61\)
Therefore, the measure of Angle MOP is **61°**.
2. **Identifying the format of proof:**
The flowchart has boxes that lead to a conclusion through definitions and given statements. Because of this format, the proof is a **flowchart proof**.
3. **Identifying the line segment in the figure:**
The vertical line extending through point x to point z, and the horizontal line through point y, suggests that W is another point on the circumference of the circle. However, since we’re interested in which line segment is drawn in the figure, and considering that:
- Line segment YZ would connect the top and bottom of the circle.
- Line segment WY would connect the left and right points.
- Line segment XZ connects the center to the bottom of the circle.
- Line segment WZ connects the left to the bottom.
Given the information, the most likely line segment described in the figure among these choices is **Line segment XZ** (the vertical line through the center to the edge).
4. **Determining the true statement about given information:**
Because point E is the vertex where the right angle is formed at CED, we know that:
- ∠CED measures 90° because it is a right angle.
- Since ∠AEC is formed by a straight horizontal line and a vertical line, that means ∠AEC also measures 90°.
Therefore, **∠AEC measures 90°**.
Summary of answers:
1. **61°**
2. **flowchart proof**
3. **Line segment XZ**
4. **∠AEC measures 90°**
1. **Finding the measure of Angle MOP:**
Since Angle MON is a straight angle, it measures 180 degrees. Ray OP bisects Angle MOQ. Therefore, we note that:
- If the angle between OQ and ON is 58 degrees, then the angle between OM and OP (let's call it x) plus the angle between OP and OQ (58 degrees) plus the angle between OQ and ON (also x since OP bisects the angle) must add up to 180 degrees.
So, we can set up the equation:
\(x + 58 + x = 180\)
Simplifying this gives:
\(2x + 58 = 180\)
\(2x = 180 - 58\)
\(2x = 122\)
\(x = 61\)
Therefore, the measure of Angle MOP is **61°**.
2. **Identifying the format of proof:**
The flowchart has boxes that lead to a conclusion through definitions and given statements. Because of this format, the proof is a **flowchart proof**.
3. **Identifying the line segment in the figure:**
The vertical line extending through point x to point z, and the horizontal line through point y, suggests that W is another point on the circumference of the circle. However, since we’re interested in which line segment is drawn in the figure, and considering that:
- Line segment YZ would connect the top and bottom of the circle.
- Line segment WY would connect the left and right points.
- Line segment XZ connects the center to the bottom of the circle.
- Line segment WZ connects the left to the bottom.
Given the information, the most likely line segment described in the figure among these choices is **Line segment XZ** (the vertical line through the center to the edge).
4. **Determining the true statement about given information:**
Because point E is the vertex where the right angle is formed at CED, we know that:
- ∠CED measures 90° because it is a right angle.
- Since ∠AEC is formed by a straight horizontal line and a vertical line, that means ∠AEC also measures 90°.
Therefore, **∠AEC measures 90°**.
Summary of answers:
1. **61°**
2. **flowchart proof**
3. **Line segment XZ**
4. **∠AEC measures 90°**