Question

g.gsr.3.4 (mc)
use the diagram to find the measure of
∠dce? what property did you use to
find the angle measure?
diagram of two triangles forming an hourglass shape with intersection at point c, angle at c in triangle abc is 58°
○ 122°; supplementary angles
○ 58°; vertical angles theorem
○ 122°; linear pair postulate
○ 58°; corresponding angle theorem

Explanation:

Step1: Identify Angle Relationship

∠DCE and the 58° angle form a linear pair (they are adjacent and form a straight line), so they are supplementary? Wait, no—wait, actually, ∠DCE and the angle adjacent to the 58° angle (wait, no, looking at the diagram: the intersection at C, so ∠ACB is 58°, and ∠DCE is vertical to... Wait, no, maybe I misread. Wait, the diagram is two triangles intersecting at C, so ∠ACB (58°) and ∠DCE: are they vertical angles? Wait, no, vertical angles are opposite each other. Wait, no, ∠ACB and ∠DCE: wait, no, the angle adjacent to ∠DCE would be the 58° angle's linear pair? Wait, no, let's re-examine.

Wait, the two triangles are A-B-C and D-E-C, intersecting at C. So ∠ACB is 58°, and ∠DCE: are they vertical angles? Wait, no, vertical angles are formed by two intersecting lines. So lines AB and DE? No, lines AC and EC, BC and DC? Wait, the intersection is at C, so the two lines are AC-E and BC-D? Wait, no, the diagram is like an hourglass: A connected to B and C, D connected to E and C, with C being the intersection. So ∠ACB (58°) and ∠DCE: are they vertical angles? Wait, no, vertical angles are opposite, so ∠ACB and ∠DCE—wait, no, ∠ACB and ∠DCE: wait, maybe ∠ACB and ∠DCE are vertical angles? Wait, no, vertical angles are equal. Wait, but the options: let's check the options.

Wait, the options include "58°; Vertical Angles Theorem" and "122°; Linear Pair Postulate". Wait, let's think again. The angle given is 58° (∠ACB). ∠DCE: if we consider the linear pair, ∠ACB and ∠DCE—wait, no, ∠ACB and ∠DCE: are they adjacent? Wait, no, the lines are AC and DC, BC and EC? Wait, maybe the angle adjacent to ∠DCE is the 58° angle's supplementary angle? Wait, no, let's recall: linear pair postulate says that two angles forming a linear pair are supplementary (sum to 180°). Wait, no, wait: ∠ACB is 58°, and ∠DCE: are they vertical angles? Wait, no, vertical angles are opposite, so ∠ACB and ∠DCE—wait, maybe I made a mistake. Wait, the correct approach: when two lines intersect, vertical angles are equal, and linear pairs are supplementary.

Wait, looking at the diagram: the two triangles intersect at C, so the angle at C for triangle ABC is 58°, and the angle at C for triangle DEC is ∠DCE. Now, are ∠ACB and ∠DCE vertical angles? If so, they should be equal (58°), but let's check the options. Wait, but the other option is 122° with linear pair. Wait, maybe I misidentified the angle. Wait, maybe the angle adjacent to ∠DCE is the 58° angle, forming a linear pair. So ∠DCE + 58° = 180°, so ∠DCE = 180 - 58 = 122°? But that would be linear pair postulate. Wait, but the options: let's check the options again.

Options:

1. 122°; Supplementary Angles

2. 58°; Vertical Angles Theorem

3. 122°; Linear Pair Postulate

4. 58°; Corresponding Angle Theorem

Wait, corresponding angles are for parallel lines, so that's out. Vertical angles: if ∠ACB and ∠DCE are vertical angles, they should be equal (58°), but is that correct? Wait, no, vertical angles are formed by two intersecting lines. So lines AC and EC, BC and DC? Wait, no, the two lines are AC and DC, BC and EC? Wait, no, the intersection is at C, so the two lines are AC-E and BC-D. So the four angles at C are ∠ACB (58°), ∠BCD, ∠DCE, and ∠ECA. So ∠ACB and ∠DCE: are they vertical angles? Yes! Because they are opposite each other when two lines intersect. Wait, no, vertical angles are opposite, so ∠ACB and ∠DCE are vertical angles, so they should be equal (58°). But wait, the other option is 122° with linear pair. Wait, maybe I got the angle wrong. Wait, maybe the angle given is ∠ACB = 58°, and ∠DCE is adjacent to it, forming a linear pair. Wait, no, let's look at the diagram again. The diagram shows ∠ACB as 58°, and ∠DCE is the angle above the intersection. Wait, maybe the angle adjacent to ∠DCE is the 58° angle, so they are supplementary. Wait, 180 - 58 = 122°, so ∠DCE = 122°, using linear pair postulate. But why?

Wait, let's clarify: when two lines intersect, they form two pairs of vertical angles (equal) and two pairs of linear pairs (supplementary). So if ∠ACB is 58°, then its vertical angle (opposite) is ∠DCE? No, wait, no: ∠ACB and ∠DCE—wait, no, ∠ACB and ∠DCE: if the lines are AC and EC, BC and DC, then the vertical angles would be ∠ACB and ∠DCE, and ∠BCD and ∠ACE. So ∠ACB and ∠DCE are vertical angles, so they should be equal (58°). But the options have "58°; Vertical Angles Theorem" as an option. But wait, the other option is "122°; Linear Pair Postulate". Which is correct?

Wait, maybe the diagram is different. Let me re-express: the triangle ABC has angle at C as 58°, and triangle DEC has angle at C as ∠DCE. If the lines AC and DC are a straight line? No, that can't be. Wait, no, the diagram is two triangles: A connected to B and C, D connected to E and C, with C being the intersection. So the sides are AC, BC, DC, EC. So the two lines are AC-EC and BC-DC, intersecting at C. So ∠ACB (58°) and ∠DCE: are they vertical angles? Yes, because they are opposite each other. So vertical angles are equal, so ∠DCE = 58°, using Vertical Angles Theorem. But wait, the other option is 122° with linear pair. Wait, maybe I misread the angle. Wait, maybe the angle given is ∠ACD = 58°, not ∠ACB. Wait, the problem says "the diagram to find the measure of ∠DCE". The diagram shows 58° at ∠ACB. Wait, maybe the angle adjacent to ∠DCE is 58°, so they are supplementary. Wait, 180 - 58 = 122°, so ∠DCE = 122°, using linear pair postulate. But why the confusion?

Wait, let's check the options:

Option 2: 58°; Vertical Angles Theorem. If ∠ACB and ∠DCE are vertical angles, then they are equal, so 58°, which matches. But wait, the linear pair postulate: if ∠ACB and ∠DCE are adjacent and form a linear pair, then they sum to 180°. But in the diagram, are they adjacent? Let's see: the two triangles are A-B-C and D-E-C, intersecting at C. So ∠ACB is at the bottom triangle, and ∠DCE is at the top triangle. So the lines are AC and EC (forming a straight line?), and BC and DC (forming a straight line?). So ∠ACB (58°) and ∠DCE: are they vertical angles? Yes, because they are opposite each other when two lines (AC-EC and BC-DC) intersect. So vertical angles are equal, so ∠DCE = 58°, using Vertical Angles Theorem. So the correct option is the second one: 58°; Vertical Angles Theorem.

Wait, but let's confirm. Vertical Angles Theorem states that vertical angles are congruent (equal). So if ∠ACB and ∠DCE are vertical angles, then they are equal. So ∠DCE = 58°, using Vertical Angles Theorem. So the correct option is "58°; Vertical Angles Theorem".

Step2: Evaluate Options

- Option 1: 122°; Supplementary Angles. Supplementary angles sum to 180°, but ∠DCE and 58° would need to be supplementary, but they are vertical angles, not supplementary. So incorrect.
- Option 2: 58°; Vertical Angles Theorem. Vertical angles are equal, so ∠DCE = 58°, correct.
- Option 3: 122°; Linear Pair Postulate. Linear pair sums to 180°, but ∠DCE and 58° are not a linear pair (they are vertical angles), so incorrect.
- Option 4: 58°; Corresponding Angle Theorem. Corresponding angles are for parallel lines, not applicable here. Incorrect.

Answer:

B. 58°; Vertical Angles Theorem (assuming the options are labeled as A, B, C, D with B being "58°; Vertical Angles Theorem")