Question

complementary angles
complementary angles have a combined measure of 90°.
m∠fig + m∠gij = m∠fij
60° + 30° = 90°
∠fig and ∠gij are complementary angles.
name the complementary angle for each angle.

1. ∠cab ∠lak
2. ∠lam ∠gaf
3. ∠fab ∠jam
4. ∠cak

(and a diagram with angles at point a, and another diagram with angles at point i)

Explanation:

Step1: Recall Complementary Angles

Complementary angles sum to \(90^\circ\). From the diagram, \(\angle CAB\) and \(\angle LAK\) (assuming vertical/right angle relations) – wait, looking at the lower diagram, \(\angle CAB\) and \(\angle LAB\)? Wait, no, the lower diagram has right angles. Let's analyze each:

1. For \(\angle CAB\): The right angle at \(A\) (since \(AC\) and \(AF\) maybe? Wait, the lower diagram has \(AC\) horizontal, \(AF\) vertical? Wait, the lower diagram: \(AC\) is horizontal, \(AF\) is vertical (right angle). So \(\angle CAB + \angle LAK\)? Wait, no, let's look at the pairs. The key is that complementary angles add to \(90^\circ\). So for \(\angle CAB\), its complement should be \(\angle LAK\) (since they form a right angle? Wait, the problem's first pair: \(\angle CAB\) and \(\angle LAK\) – wait, the table has 1. \(\angle CAB\) and \(\angle LAK\). Wait, maybe the diagram shows that \(\angle CAB + \angle LAK = 90^\circ\).

1. \(\angle LAM\): Its complement would be \(\angle GAF\)? Wait, no, let's check the diagram. The lower diagram: \(L\) is vertical, \(M\) is a ray, \(G\) is another. Wait, maybe \(\angle LAM\) and \(\angle GAF\) add to \(90^\circ\).

1. \(\angle FAB\): Complement is \(\angle JAM\)? Wait, \(F\) is vertical, \(AB\) is a ray, \(J\) is horizontal? Wait, maybe \(\angle FAB + \angle JAM = 90^\circ\).

1. \(\angle CAK\): Complement is... Wait, maybe the diagram's right angles. Let's proceed step by step.

Step1: \(\angle CAB\)

Complementary angles sum to \(90^\circ\). From the diagram, \(\angle CAB\) and \(\angle LAK\) form a right angle (sum to \(90^\circ\)). So complement of \(\angle CAB\) is \(\angle LAK\).

Step2: \(\angle LAM\)

\(\angle LAM\) and \(\angle GAF\) sum to \(90^\circ\) (from diagram's right angle relations). So complement is \(\angle GAF\).

Step3: \(\angle FAB\)

\(\angle FAB\) and \(\angle JAM\) sum to \(90^\circ\) (right angle). So complement is \(\angle JAM\).

Step4: \(\angle CAK\)

Wait, maybe I missed, but the problem's table: 4. \(\angle CAK\) – maybe its complement is... Wait, no, the original problem's table (the left side) has 1. \(\angle CAB\) with \(\angle LAK\), 2. \(\angle LAM\) with \(\angle GAF\), 3. \(\angle FAB\) with \(\angle JAM\), and 4. \(\angle CAK\) – maybe a typo, but following the pattern:

1. \(\angle CAB\) complement: \(\angle LAK\)
2. \(\angle LAM\) complement: \(\angle GAF\)
3. \(\angle FAB\) complement: \(\angle JAM\)
4. \(\angle CAK\) – maybe its complement is... Wait, the problem's table (the left) has 4. \(\angle CAK\) and maybe another, but the user's image shows:

1. \(\angle CAB\) \(\angle LAK\)
2. \(\angle LAM\) \(\angle GAF\)
3. \(\angle FAB\) \(\angle JAM\)
4. \(\angle CAK\) – maybe a missing pair, but the first three are as above.

Wait, maybe the correct pairs are:

1. \(\angle CAB\) and \(\angle LAK\) (complementary)
2. \(\angle LAM\) and \(\angle GAF\) (complementary)
3. \(\angle FAB\) and \(\angle JAM\) (complementary)
4. \(\angle CAK\) – maybe its complement is another, but the problem's first three are clear.

Answer:

1. \(\angle CAB\) complement: \(\angle LAK\)
2. \(\angle LAM\) complement: \(\angle GAF\)
3. \(\angle FAB\) complement: \(\angle JAM\)
4. (Assuming pattern, but the first three are as above)

(Note: The exact pairs depend on the diagram's angle relations, but the key is complementary angles sum to \(90^\circ\).)