Free Angle Pair Relationships Worksheets For Teaching - Free Printable

Let's solve each part step by step, using angle relationships like vertical angles, linear pairs (supplementary angles), and right angles.

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## 🔹 Problem 1:

We have point G as the vertex. Lines AC (horizontal), EF (vertical), and rays GD and GB are drawn.

Given:
- ∠DGC = 40°
- ∠FGE = 45°
- Since EF is vertical and AC is horizontal, they are perpendicular → ∠EGC = 90°, ∠EGA = 90°, etc.

i) m∠DGE = ?

∠DGE is the angle between ray GD and ray GE.

We know:
- ∠EGC = 90° (since EF ⊥ AC)
- ∠DGC = 40°

So, ∠DGE = ∠EGC - ∠DGC = 90° - 40° = 50°

✔ Answer: 50°

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ii) m∠AGF = ?

∠AGF is the angle between ray GA and ray GF.

We know:
- ∠FGE = 45°
- ∠EGA = 90° (since EF ⊥ AC)

So, ∠AGF = ∠EGA - ∠FGE = 90° - 45° = 45°

✔ Answer: 45°

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iii) m∠CGE = ?

This is the angle between ray GC and ray GE.

Since EF is vertical and AC is horizontal, they are perpendicular → ∠CGE = 90°

✔ Answer: 90°

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iv) m∠CGB = ?

∠CGB is the angle between ray GC and ray GB.

Notice that ray GB is the extension of ray GD (opposite direction). So, ∠CGB is the vertical angle to ∠DGA.

But easier: since ∠DGC = 40°, and points D, G, B are colinear (straight line), then ∠CGB is supplementary to ∠DGC.

Wait — actually, if D-G-B is a straight line, then ∠DGC + ∠CGB = 180°? No — because D and B are on opposite sides of G.

Actually, looking at the diagram: Ray GD and ray GB are opposite rays → so line DB passes through G.

Therefore, ∠DGC and ∠CGB are adjacent angles forming a straight line → they are supplementary.

So, ∠CGB = 180° - ∠DGC = 180° - 40° = 140°

✔ Answer: 140°

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v) m∠AGB = ?

∠AGB is the angle from ray GA to ray GB.

We can think of it as:

From GA to GC is 180° (straight line).

From GC to GB is ∠CGB = 140° (from above).

Wait — better way:

Since GA and GC are opposite rays → ∠AGC = 180°

∠AGB = ∠AGC - ∠CGB = 180° - 140° = 40°

Alternatively, notice that ∠AGB and ∠DGC are vertical angles → they are equal!

Yes! Because lines AC and DB intersect at G → vertical angles: ∠AGB = ∠DGC = 40°

✔ Answer: 40°

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## ✔ Final Answers for Problem 1:

i) m∠DGE = 50°
ii) m∠AGF = 45°
iii) m∠CGE = 90°
iv) m∠CGB = 140°
v) m∠AGB = 40°

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## 🔹 Problem 2:

Lines VT and PS are horizontal, intersected by transversal UR at points W and X respectively.

Given:
- ∠SXR = 79°

We’ll use:
- Vertical angles
- Linear pairs (supplementary)
- Corresponding/alternate angles if lines are parallel — but not stated. However, since both VT and PS are horizontal, we assume they are parallel.

i) m∠VWU = ?

∠VWU is at point W, between rays WV and WU.

Note: ∠VWU and ∠SXR are corresponding angles if VT ∥ PS and UR is transversal.

Since VT and PS are both horizontal → parallel.

So, ∠VWU = ∠SXR = 79° (corresponding angles)

✔ Answer: 79°

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ii) m∠PXR = ?

∠PXR is at point X, between rays XP and XR.

XP and XS are opposite rays → ∠PXS = 180°

Given: ∠SXR = 79°

So, ∠PXR = 180° - ∠SXR = 180° - 79° = 101°

✔ Answer: 101°

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iii) m∠PXW = ?

∠PXW is at point X, between rays XP and XW.

Note: Points W and X are on the same transversal UR.

Rays XW and XR are opposite rays? Let’s see: U-W-X-R — so W and R are on opposite sides of X → so ray XW and ray XR are opposite → line WR passes through X.

Thus, ∠PXW and ∠PXR are adjacent angles forming a straight line? Wait — no.

Actually, ∠PXW is the angle between XP and XW.

Since ∠PXR = 101° (from above), and ∠PXW and ∠PXR are adjacent angles sharing ray XP, and rays XW and XR are opposite → so ∠PXW + ∠PXR = 180°?

Wait — yes! Because XW and XR form a straight line → so ∠WX R = 180°.

Therefore, ∠PXW = 180° - ∠PXR = 180° - 101° = 79°

Alternatively, ∠PXW and ∠SXR are vertical angles? Let’s check:

At point X, rays XP, XS (opposite), and XR, XW (opposite).

So, ∠PXW and ∠SXR are vertical angles → equal → 79°

✔ Answer: 79°

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iv) m∠UWT = ?

∠UWT is at point W, between rays WU and WT.

WT is along the horizontal line VT.

We know ∠VWU = 79° (from part i)

Since VW and WT are opposite rays → ∠VWT = 180°

So, ∠UWT = 180° - ∠VWU = 180° - 79° = 101°

✔ Answer: 101°

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v) m∠WXS = ?

∠WXS is at point X, between rays XW and XS.

We know ∠SXR = 79°, and ∠WXS and ∠SXR are adjacent angles sharing ray XS, and rays XW and XR are opposite → so they form a linear pair.

Thus, ∠WXS + ∠SXR = 180° → ∠WXS = 180° - 79° = 101°

✔ Answer: 101°

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## ✔ Final Answers for Problem 2:

i) m∠VWU = 79°
ii) m∠PXR = 101°
iii) m∠PXW = 79°
iv) m∠UWT = 101°
v) m∠WXS = 101°

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## 📌 Summary:

Problem 1:

i) 50°
ii) 45°
iii) 90°
iv) 140°
v) 40°

Problem 2:

i) 79°
ii) 101°
iii) 79°
iv) 101°
v) 101°

All answers are based on fundamental angle relationships: vertical angles, linear pairs, perpendicular lines, and corresponding angles (assuming parallel lines where appropriate).