Three lines are shown. A line with points C, A, and F intersects a line with points E, A, and B at point A. A line extends from point A to point D in between angle EAF. Angle CAE is 61 degrees, angle DAF is 90 degrees, and angle FAB is 61 degrees. Find the following angle measures. mAngleEAD = ° mAngleCAB = °
The Correct Answer and Explanation is:
We are given the following geometric configuration:
- Line CAF (with points C, A, and F) intersects line EAB (with points E, A, and B) at point A.
- A third line extends from A to D, lying between angle EAF.
- ∠CAE = 61°
- ∠DAF = 90°
- ∠FAB = 61°
We are asked to find:
- m∠EAD = ?
- m∠CAB = ?
Step-by-Step Solution:
We know:
- ∠CAE = 61°
- ∠DAF = 90°
- ∠FAB = 61°
- Line AD lies between the rays AE and AF, meaning angle ∠EAF is split by ray AD into ∠EAD and ∠DAF.
- Similarly, line AB lies on the opposite side of ray AF, such that ∠FAB is adjacent to ∠DAF.
Let’s break this down:
1. Find m∠EAD
From the given:
- ∠EAF is made up of ∠EAD + ∠DAF.
- ∠DAF = 90°
- ∠CAE = 61°, and since point C is on the same side of A as E, this tells us ∠CAE = ∠EAF = 61°.
So,
∠EAD = ∠EAF − ∠DAF
= 61° − 90°
= Not possible.
This tells us that ∠CAE must actually refer to a different angle—not ∠EAF.
Let’s re-interpret the configuration more carefully.
Better Interpretation of Given Angles:
Let’s draw a mental image:
- C——A——F is a straight line.
- E——A——B is a second line crossing at A.
- D lies between ∠EAF, forming two smaller angles: ∠EAD and ∠DAF = 90°.
Given:
- ∠CAE = 61°
→ This is the angle between ray AC and ray AE. - ∠FAB = 61°
→ This is the angle between ray AF and ray AB. - ∠DAF = 90°
→ So ray AD is perpendicular to ray AF.
We are to find:
- ∠EAD
- ∠CAB
Find m∠EAD:
We know:
- ∠CAE = 61°, and line CAF is straight, so angle between AE and AD (∠EAD) can be found from:
∠EAD = ∠EAF − ∠DAF
∠EAF = ∠CAE (since AE and AC are on same side) = 61°
∠DAF = 90°
So:
∠EAD = 61° − 90° = −29°, which is impossible.
So instead, use this logic:
∠EAD = ∠DAF − ∠FAE
But that requires knowing ∠FAE, which we don’t.
However, we can treat ∠EAD and ∠DAF as two angles that sum to ∠EAF. So:
∠EAD = ∠EAF − ∠DAF = 61° − 90° = again −29°
Again, no good.
But if ∠DAF = 90°, and ∠FAB = 61°, then ∠DAB = ∠DAF + ∠FAB = 90° + 61° = 151°
Similarly, ∠EAD = ∠CAE − ∠CAD
But without ∠CAD, we can’t use that.
Let’s instead focus on full straight angle:
Since CAF is a line, angle ∠CAB is a straight angle:
∠CAB = ∠CAE + ∠EAD + ∠DAF + ∠FAB
So:
∠CAB = 61° (CAE) + ∠EAD + 90° + 61° (FAB)
But this is incorrect—∠CAE and ∠FAB are both parts of ∠CAB already.
Better approach:
Use Linear Angle:
CAF is a straight line → angle between ray AC and ray AF = 180°
So the three adjacent angles ∠CAE, ∠EAD, and ∠DAF must sum to 180°:
- ∠CAE = 61°
- ∠DAF = 90°
- So ∠EAD = 180° − 61° − 90° = 29°
✅ m∠EAD = 29°
Now, find m∠CAB
This is the angle between ray AC and ray AB.
Given:
- ∠DAF = 90°
- ∠FAB = 61°
So:
∠CAB = ∠DAF + ∠FAB = 90° + 61° = 151°
✅ m∠CAB = 151°
Final Answers:
- m∠EAD = 29°
- m∠CAB = 151°
Explanation
This geometry problem presents three intersecting lines with given angle measures. The key to solving it lies in understanding how these angles are arranged around point A. The line CAF is straight, which means that the total angle along this line is 180°. Points C, A, and F lie on this straight line. We are also told that ∠CAE = 61°, ∠DAF = 90°, and ∠FAB = 61°.
To find m∠EAD, we notice that angle ∠CAE, ∠EAD, and ∠DAF are adjacent angles along the straight line CAF. Since a straight line measures 180°, the sum of these three angles must equal 180°. Using the given values, we calculate:
∠EAD = 180° − ∠CAE − ∠DAF = 180° − 61° − 90° = 29°.
Next, to find m∠CAB, we observe that it spans from ray AC to ray AB, passing through ray AD and ray AF. The angle ∠CAB is the sum of the angles ∠DAF (between AD and AF) and ∠FAB (between AF and AB). Since both values are provided (90° and 61°), we simply add:
∠CAB = ∠DAF + ∠FAB = 90° + 61° = 151°.
This solution highlights how adjacent angles along a straight line or sharing a vertex can be used to determine unknown angle measures. By recognizing the relationship between the angles, especially how they compose a straight angle or form larger composite angles, we can apply basic angle addition and subtraction to solve for unknowns accurately.
