Lesson 7 3 angle relationships | PDF - Free Printable

You're working on Lesson 7-3: Angle Relationships, and the image shows two intersecting lines forming four angles around point B, with given angle measures: 130°, 50°, 50°, and 30°. You’re also given a vocabulary table defining key angle relationships.

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## 🔍 STEP 1: Understand the Diagram

The diagram shows two lines intersecting at point B, forming four angles:

- ∠ABC = 130°
- ∠CBF = 50°
- ∠FBD = 30° (Wait — this seems inconsistent!)
- ∠DBA = 50°

But here’s the issue: If two straight lines intersect, the angles around the point should add up to 360°, and opposite angles (vertical angles) should be equal.

Let’s check the given angles:

> 130° + 50° + 30° + 50° = 260° → ✘ Not 360° → This is impossible for intersecting lines!

So there’s likely a typo or mislabeling in the diagram.

Looking more carefully, the 30° label is probably meant to be 50°, because:

> 130° + 50° + 130° + 50° = 360° ✔
And vertical angles would be equal: 130° opposite 130°, 50° opposite 50°.

✔ So we’ll assume the 30° is a mistake and should be 50°. This makes sense geometrically and matches the vocabulary examples given (which list ∠ABC and ∠DBF as vertical angles — both should be 130° — and ∠CBF and ∠ABD as vertical angles — both should be 50°).

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## 📚 STEP 2: Apply Vocabulary Definitions

Let’s use the definitions from your table to identify relationships.

1. Congruent Angles

> “When two angles have the same measure.”

✔ ∠ABC = 130° and ∠DBF = 130° → Congruent
✔ ∠CBF = 50° and ∠ABD = 50° → Congruent

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2. Vertical Angles

> “Are formed opposite each other when two lines intersect. They have the same measure and share a vertex.”

✔ ∠ABC and ∠DBF are vertical → both 130°
✔ ∠CBF and ∠ABD are vertical → both 50°

*(Note: The diagram incorrectly labels one angle as 30° — it should be 50° to match vertical angle pairs.)*

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3. Adjacent Angles

> “Are side by side and have a common vertex and ray. They may or may not be congruent.”

Examples from table:
- ∠ABC and ∠CBF → share ray BC → adjacent ✔
- ∠CBF and ∠FBD → if FBD were 50°, they’d be adjacent, but labeled 30° → inconsistency.
- Better to stick with correct pairs: ∠CBF and ∠ABD? No — they don’t share a ray.

Actually, correct adjacent pairs sharing vertex B:
- ∠ABC & ∠CBF (share ray BC)
- ∠CBF & ∠FBD (share ray BF) — but again, 30° is wrong
- ∠FBD & ∠DBA (share ray BD)
- ∠DBA & ∠ABC (share ray BA)

Assuming all angles are correctly labeled as 130°, 50°, 130°, 50° going around:

Then adjacent pairs are:
- 130° & 50°
- 50° & 130°
- 130° & 50°
- 50° & 130°

All adjacent pairs are not congruent (130 ≠ 50), which is fine — definition says “may or may not be congruent.”

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4. Complementary Angles

> “Two angles whose measure have a sum of 90°”

Check all pairs:
- 130° + 50° = 180° → not complementary
- 50° + 50° = 100° → no
- 130° + 130° = 260° → no

✘ No complementary angles in this diagram.

*(The example in the table shows a different diagram with 60° and 30° — that’s separate.)*

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5. Supplementary Angles

> “Two angles whose measure have a sum of 180°”

Check pairs:
- ∠ABC (130°) + ∠CBF (50°) = 180° → ✔ supplementary
- ∠CBF (50°) + ∠FBD (should be 130°) = 180° → ✔
- ∠FBD (130°) + ∠DBA (50°) = 180° → ✔
- ∠DBA (50°) + ∠ABC (130°) = 180° → ✔

Also, any two adjacent angles that form a straight line are supplementary.

✔ So every pair of adjacent angles here is supplementary.

Also, vertical angles are NOT supplementary unless they are 90° — here, 130° and 50° are not supplementary to themselves.

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## ✔ FINAL ANSWERS / SUMMARY

| Relationship | Pairs |
|---------------------|-----------------------------------------------------------------------|
| Congruent | ∠ABC ≅ ∠DBF (130°), ∠CBF ≅ ∠ABD (50°) |
| Vertical Angles | ∠ABC & ∠DBF; ∠CBF & ∠ABD |
| Adjacent Angles | ∠ABC & ∠CBF; ∠CBF & ∠FBD; ∠FBD & ∠DBA; ∠DBA & ∠ABC |
| Complementary | None (no pair adds to 90°) |
| Supplementary | All adjacent pairs: e.g., ∠ABC + ∠CBF = 180°, etc. |

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## 🛠️ CORRECTION NOTE

There is an error in the diagram: The angle labeled 30° should be 50° to make the geometry valid (sum to 360°, vertical angles equal). If you’re submitting this assignment, you might want to note that correction or ask your teacher about it.

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## 💡 KEY TAKEAWAY

When two lines intersect:
- Vertical angles are equal.
- Adjacent angles are supplementary (add to 180°).
- The total around the point is 360°.

Always double-check that the given angles follow these rules — if not, there’s likely a typo!

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✔ You’ve now solved the problem and understood how to apply angle relationship definitions to a diagram — even when the diagram has an error!