Question
Lines a and b in the diagram are parallel.
Parallel lines a and b are cut by 2 transversals. At the intersection of line a with the transversals, clockwise from top left, the angles are 45 degrees, 95 degrees, blank, blank, B, 40 degrees. At the intersection of b with a transversal, the angles are blank, A, blank, blank. At the intersection of b with the second transversal, the angles are C, blank, blank, blank.
Step 1: measure of angle A = 40 degrees (alternate interior angles). Step 2: measure of angle B = 95 degrees (vertical angles). Step 3: measure of angle C = 40 degrees (corresponding angles).
Shalini found the sum of the measures of the interior angles of the triangle to be 175°. She knows the sum is supposed to be 180°. What was Shalini’s error?
m∠A = 45°
m∠B = 85°
m∠C = 45°
Shalini did not add the sum of the 3 angles correctly.
Parallel lines a and b are cut by 2 transversals. At the intersection of line a with the transversals, clockwise from top left, the angles are 45 degrees, 95 degrees, blank, blank, B, 40 degrees. At the intersection of b with a transversal, the angles are blank, A, blank, blank. At the intersection of b with the second transversal, the angles are C, blank, blank, blank.
Step 1: measure of angle A = 40 degrees (alternate interior angles). Step 2: measure of angle B = 95 degrees (vertical angles). Step 3: measure of angle C = 40 degrees (corresponding angles).
Shalini found the sum of the measures of the interior angles of the triangle to be 175°. She knows the sum is supposed to be 180°. What was Shalini’s error?
m∠A = 45°
m∠B = 85°
m∠C = 45°
Shalini did not add the sum of the 3 angles correctly.
Answers
GPT-4o mini
It appears that Shalini made an error in determining the measure of the angles A, B, and C. Let's go through the angles again based on the provided information.
1. **Measure of angle A:**
From step 1, we have \( \text{m} \angle A = 40^\circ \) (correctly identified as alternate interior angles).
2. **Measure of angle B:**
From step 2, we derived \( \text{m} \angle B = 95^\circ \) (correctly identified as vertical angles).
3. **Measure of angle C:**
From step 3, we derived \( \text{m} \angle C = 40^\circ \) (identified as corresponding angles with the same transversal).
Now, let's clarify the angles again:
- From the information provided, the angles around line a are:
- 45 degrees
- 95 degrees
- (blank)
- (blank)
- B (given as 95 degrees)
- 40 degrees
Since the angle \( B \) is 95 degrees, the adjacent angle to \( B \) (in a linear pair) would be \( 180^\circ - 95^\circ = 85^\circ \). Therefore, it appears that Shalini has made mistakes in identifying the measure of angle B.
This analysis indicates:
- **m∠A = 40°** (correct)
- **m∠B = 95°** (correct)
- The angle adjacent to \( B \) should lead to the conclusion that the third angle (which seems to be listed incorrectly earlier as \( C \)) should be calculated correctly.
If we consider Shalini's earlier statement of summing up the angles:
1. **When she states:**
- \( m∠A = 45° \) (this is incorrect)
- \( m∠B = 85° \) (if she thinks this is the external angle to B, then she could mistake)
- \( m∠C = 45° \) (this needs to be rebuilt from the angle analyses).
Ultimately, the correct angles while validating Shalini's triangle should have summed to:
1. 40°
2. 95°
3. 45°
But since angle C should not count with \( 40^\circ \) (as previously determined) - thus leading us back to:
The correct configuration is:
- 65° etc, which all must total up again to check over.
Conclusion: Shalini misunderstood the sum because of misinterpretation of angle completeness. Therefore, her angles should indeed be corrected to add to an expected sum of 180°, rather than the wrongly computed sum of 175°.
1. **Measure of angle A:**
From step 1, we have \( \text{m} \angle A = 40^\circ \) (correctly identified as alternate interior angles).
2. **Measure of angle B:**
From step 2, we derived \( \text{m} \angle B = 95^\circ \) (correctly identified as vertical angles).
3. **Measure of angle C:**
From step 3, we derived \( \text{m} \angle C = 40^\circ \) (identified as corresponding angles with the same transversal).
Now, let's clarify the angles again:
- From the information provided, the angles around line a are:
- 45 degrees
- 95 degrees
- (blank)
- (blank)
- B (given as 95 degrees)
- 40 degrees
Since the angle \( B \) is 95 degrees, the adjacent angle to \( B \) (in a linear pair) would be \( 180^\circ - 95^\circ = 85^\circ \). Therefore, it appears that Shalini has made mistakes in identifying the measure of angle B.
This analysis indicates:
- **m∠A = 40°** (correct)
- **m∠B = 95°** (correct)
- The angle adjacent to \( B \) should lead to the conclusion that the third angle (which seems to be listed incorrectly earlier as \( C \)) should be calculated correctly.
If we consider Shalini's earlier statement of summing up the angles:
1. **When she states:**
- \( m∠A = 45° \) (this is incorrect)
- \( m∠B = 85° \) (if she thinks this is the external angle to B, then she could mistake)
- \( m∠C = 45° \) (this needs to be rebuilt from the angle analyses).
Ultimately, the correct angles while validating Shalini's triangle should have summed to:
1. 40°
2. 95°
3. 45°
But since angle C should not count with \( 40^\circ \) (as previously determined) - thus leading us back to:
The correct configuration is:
- 65° etc, which all must total up again to check over.
Conclusion: Shalini misunderstood the sum because of misinterpretation of angle completeness. Therefore, her angles should indeed be corrected to add to an expected sum of 180°, rather than the wrongly computed sum of 175°.