Geometry Module 7 Skill 8 Day 1: Isosceles and Non Isosceles ... - Free Printable

Let’s solve each problem step by step.

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Problem 5: DEFG is an isosceles trapezoid.

In an isosceles trapezoid:
- The non-parallel sides (legs) are congruent → so DG ≅ EF
- The diagonals are congruent → so DF ≅ GE

✔ So the answers are:
- DG ≅ EF
- DF ≅ GE

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Problem 6: TUVW is an isosceles trapezoid.

In an isosceles trapezoid:
- Base angles are congruent.
- Angles at each base are equal.

Looking at the diagram:
- ∠T and ∠U are on the top base → they are congruent → ∠T ≅ ∠U
- ∠V and W are on the bottom base → they are congruent → ∠V ≅ ∠W

✔ So the answers are:
- ∠T ≅ ∠U
- ∠V ≅ ∠W

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Problem 7: Trapezoid PQRS with angles given.

We’re told:
- m∠P = 112°
- m∠Q = 112°
- Angles at S and R are both marked as “x” → so m∠S = x, m∠R = x

Sum of interior angles in any quadrilateral = 360°

So:
112 + 112 + x + x = 360
→ 224 + 2x = 360
→ 2x = 360 - 224 = 136
→ x = 68

So:
- m∠Q = 112° (already given)
- m∠R = 68°
- m∠S = 68°

✔ Final answers for #7:
- m∠Q = 112°
- m∠R = 68°
- m∠S = 68°

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Problem 8: Trapezoid WXYZ with one angle given.

Given: m∠X = 47°

Assuming this is also an isosceles trapezoid (since it's grouped with others and has parallel sides shown), then:

- Base angles are congruent.
- Since WX and YZ are the bases (arrows show they’re parallel), then:
- ∠W and ∠X are on the same base? Wait — let’s look carefully.

Actually, looking at the arrows:
- Side WX has arrow → side YZ has arrow → so WX || YZ → those are the two bases.
- Then legs are WZ and XY.

In an isosceles trapezoid with bases WX and YZ:
- Angles adjacent to each leg are supplementary (because consecutive angles between parallel lines are supplementary).
- Also, base angles are congruent: ∠W ≅ ∠X? No — wait.

Actually, if WX and YZ are the bases, then:
- ∠W and Z are adjacent to leg WZ → not necessarily equal.
Wait — better approach:

In trapezoid WXYZ with WX || YZ:
- Consecutive angles between the parallel sides are supplementary.
So:
- ∠W + Z = 180°
- ∠X + ∠Y = 180°

But we’re told it’s likely isosceles (from context of worksheet). In isosceles trapezoid:
- Base angles are congruent → so ∠W ≅ ∠X and ∠Z ≅ ∠Y? Not quite.

Wait — standard rule: In isosceles trapezoid, each pair of base angles is congruent.

If WX and YZ are the bases, then:
- Angles at base WX: ∠W and ∠X → should be congruent? But that would mean ∠W = 47°, but then ∠Z and ∠Y would be 133° each.

But let’s check sum: 47 + 47 + 133 + 133 = 360 → yes.

Alternatively, maybe ∠X and ∠Y are base angles? Let’s think visually.

Actually, in most diagrams like this, if WX is top base and YZ is bottom base, then:
- ∠W and X are top angles → if isosceles, they are equal? No — actually, in isosceles trapezoid, the base angles are the ones on the same base.

Standard definition: In isosceles trapezoid, the angles adjacent to each leg are equal? No.

Correct rule: In an isosceles trapezoid, the base angles are congruent. That means:
- The two angles on the top base are congruent to each other? No — actually, no.

Wait — correction:

In an isosceles trapezoid, the angles on the same side of a leg are supplementary (because of parallel lines), and the base angles (angles adjacent to the same base) are congruent.

Example: If AB and CD are bases, with AB top, CD bottom, then:
- ∠A ≅ ∠B (top base angles) — NO, that’s not right.

Actually, correct property:
In isosceles trapezoid, each pair of base angles is congruent. Meaning:
- The two angles adjacent to the top base are congruent to each other? No.

I think I’m confusing myself.

Let me recall: In isosceles trapezoid, the legs are congruent, and the base angles are congruent. Specifically:
- The angles adjacent to each base are congruent.

More precisely:
- The two angles that share the same base are congruent.

For example, if bases are WX and YZ, then:
- Angles at base WX: ∠W and ∠X — these are NOT necessarily congruent unless it’s a rectangle.

Actually, no — in isosceles trapezoid, the base angles are congruent meaning:
- ∠W ∠X? Only if it’s symmetric.

Wait — let’s use the standard fact:

In isosceles trapezoid ABCD with AB || CD, then:
- ∠A ≅ ∠B and ∠C ≅ ∠D? No.

Actually, correct: ∠A ≅ ∠D and ∠B ≅ ∠C? No.

I remember now: In isosceles trapezoid, the angles on the same leg are supplementary (due to parallel lines), and the base angles are congruent — meaning the angles adjacent to the same base are congruent.

Actually, standard textbook definition: In an isosceles trapezoid, each pair of base angles is congruent. That is:
- The two angles adjacent to the top base are congruent to each other? No.

Let me think differently.

Suppose trapezoid WXYZ with WX || YZ, and it’s isosceles, so legs WZ and XY are congruent.

Then, the base angles are:
- At base WX: angles ∠W and X — but these are not necessarily equal.

Actually, the correct property is:
- The angles adjacent to each leg are supplementary (because of parallel lines).
- And because it’s isosceles, the base angles are congruent — specifically, ∠W ≅ ∠X? No.

I found my mistake.

In isosceles trapezoid with bases WX and YZ (WX top, YZ bottom), then:
- ∠W and Z are adjacent to leg WZ — not helpful.

Standard rule: In isosceles trapezoid, the base angles are congruent. This means:
- The two angles that are on the same base are congruent? No.

Actually, it’s: The angles adjacent to the same leg are not necessarily equal, but the angles on the same side of the axis of symmetry are equal.

Perhaps easier: since it’s isosceles, the left and right are mirror images.

So if ∠X = 47°, and assuming the trapezoid is oriented with WX on top, YZ on bottom, and X is top-right corner, then its mirror image is W (top-left), so ∠W = ∠X = 47°.

Then, since WX || YZ, consecutive interior angles are supplementary:
- ∠W + ∠Z = 180° → 47 + ∠Z = 180 → ∠Z = 133°
- Similarly, ∠X + ∠Y = 180° → 47 + ∠Y = 180 → ∠Y = 133°

And since it’s isosceles, ∠Z should equal ∠Y, which they do (both 133°).

Perfect.

So:
- m∠W = 47°
- m∠Y = 133°
- m∠Z = 133°

✔ Answers for #8:
- m∠W = 47°
- m∠Y = 133°
- m∠Z = 133°

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Final Answer:

Problem 5:
DG ≅ EF
DF ≅ GE

Problem 6:
∠T ≅ ∠U
∠V ≅ ∠W

Problem 7:
m∠Q = 112°
m∠R = 68°
m∠S = 68°

Problem 8:
m∠W = 47°
m∠Y = 133°
m∠Z = 133°