Question

adjust point d so the measure of angle bad is equal to the measure of angle cad. which statements are true? check all that apply.

• $overline{ab}$ is parallel to $overline{ac}$.
• $overline{ad}$ bisects $angle bac$.
• $overline{ad}$ bisects $overline{ac}$.
• $overline{ad}$ forms right angles with $overline{bc}$.
• $overline{ad}$ is the perpendicular bisector of $overline{bc}$.

Explanation:

Step1: Analyze each statement

- For "AD bisects ∠BAC": If ∠BAD = ∠CAD, then AD is the angle bisector of ∠BAC, so this is true.
- For "AD forms right angles with BC": There's no info showing AD is perpendicular to BC, so this is false.
- For "AB is parallel to AC": AB and AC meet at A, so they can't be parallel, false.
- For "AD bisects AC": AD is a segment from A to BC, not bisecting AC, false.
- For "AD is the perpendicular bisector of BC": We only know ∠BAD = ∠CAD, not that AD is perpendicular or bisects BC (BD=DC? Not given), but wait, in the diagram, if we adjust D so ∠BAD=∠CAD, and if the triangle is isoceles (maybe), but the key is angle bisector. Wait, the last option: "AD is the perpendicular bisector of BC" – no, angle bisector, not necessarily perpendicular. Wait, no, the first statement: "AD bisects ∠BAC" is true because ∠BAD=∠CAD means AD splits ∠BAC into two equal parts. Also, if in the diagram, when ∠BAD=∠CAD, and if BD=DC (maybe from the diagram's measurements, BD=18, DC=18? Wait, the left text: BD=18 mm, DC=18 mm? So D is the midpoint? And ∠BAD=∠CAD, so in a triangle, if angle bisector and median coincide, it's isoceles, and AD is also perpendicular? Wait, maybe I misread. Wait, the problem says "Adjust point D so the measure of angle BAD is equal to the measure of angle CAD". So ∠BAD=∠CAD, so AD is angle bisector (first statement true). Also, if BD=DC (from diagram, BD=18, DC=18), then AD is median, and in isoceles triangle, angle bisector, median, and altitude coincide. So then AD is perpendicular bisector? Wait, the last option: "AD is the perpendicular bisector of BC". Let's recheck:

- "AD bisects ∠BAC": True, because ∠BAD=∠CAD.
- "AD is the perpendicular bisector of BC": If BD=DC (18=18) and ∠ADB=∠ADC=90° (since angle bisector in isoceles triangle is altitude), then AD is perpendicular bisector. Wait, the second statement: "AD forms right angles with BC" – that would mean ∠ADB and ∠ADC are right angles, so if that's true, then AD is perpendicular bisector. Wait, maybe the diagram has BD=DC and when ∠BAD=∠CAD, triangle ABC is isoceles with AB=AC, so AD is angle bisector, median, and altitude. So:

1. "AD bisects ∠BAC": True (∠BAD=∠CAD).
2. "AD forms right angles with BC": True (if it's isoceles, angle bisector is altitude).
3. "AB is parallel to AC": False.
4. "AD bisects AC": False.
5. "AD is the perpendicular bisector of BC": True (bisects BC (BD=DC) and is perpendicular (right angles)).

Wait, maybe I made a mistake earlier. Let's re-express:

When ∠BAD = ∠CAD, AD is the angle bisector (first statement true). If BD = DC (from diagram, BD=18, DC=18), then D is the midpoint, so AD is median. In a triangle, if angle bisector and median coincide, the triangle is isoceles (AB=AC), so AD is also the altitude (perpendicular to BC), so AD forms right angles with BC (second statement true) and AD is the perpendicular bisector of BC (fifth statement true).

But the problem says "Check all that apply". So let's re-express each:

- "AD bisects ∠BAC": True (by definition, angle bisector).
- "AD forms right angles with BC": True (if AD is altitude, which it is in isoceles triangle with AB=AC, angle bisector = altitude).
- "AB is parallel to AC": False (intersect at A).
- "AD bisects AC": False (AD is from A to BC, not AC).
- "AD is the perpendicular bisector of BC": True (bisects BC (BD=DC) and is perpendicular (right angles)).

Wait, but the second statement: "AD forms right angles with BC" – that's saying ∠ADB and ∠ADC are right angles, so that's true. And the fifth: "AD is the perpendicular bisector of BC" – since it bisects BC (BD=DC) and is perpendicular (right angles), that's true. Also, first statement: "AD bisects ∠BAC" – true.

But maybe the diagram shows BD=DC (18 mm each), so D is midpoint, and ∠BAD=∠CAD, so triangle ABC is isoceles with AB=AC, so AD is angle bisector, median, and altitude. So:

- AD bisects ∠BAC: True.
- AD forms right angles with BC: True (altitude).
- AD is the perpendicular bisector of BC: True (bisects BC (midpoint) and is perpendicular (altitude)).

Wait, but the option "AB is parallel to AC" is false. "AD bisects AC" is false.

So the true statements are:

1. AD bisects ∠BAC.

2. AD forms right angles with BC.

5. AD is the perpendicular bisector of BC.

But let's check the options again:

Options:

1. AD bisects ∠BAC. – Check.

2. AD forms right angles with BC. – Check (if altitude).

3. AB is parallel to AC. – No.

4. AD bisects AC. – No.

5. AD is the perpendicular bisector of BC. – Check (bisects BC (BD=DC) and is perpendicular (right angles)).

But maybe the key is that when ∠BAD=∠CAD, AD is the angle bisector (first statement), and if BD=DC, then it's also perpendicular bisector. So the correct statements are "AD bisects ∠BAC", "AD forms right angles with BC", and "AD is the perpendicular bisector of BC". But let's confirm with the angle bisector definition: angle bisector splits the angle into two equal parts, so first statement is true. The last statement: perpendicular bisector means bisects the segment and is perpendicular, so if BD=DC and AD is perpendicular, then yes. The second statement: "AD forms right angles with BC" means perpendicular, so that's true.

Answer:

The true statements are:

• $\boldsymbol{\overline{AD}}$ bisects $\boldsymbol{\angle BAC}$.
• $\boldsymbol{\overline{AD}}$ forms right angles with $\boldsymbol{\overline{BC}}$.
• $\boldsymbol{\overline{AD}}$ is the perpendicular bisector of $\boldsymbol{\overline{BC}}$.

(To check the boxes: the first, second, and fifth options.)