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Back to Draw geometric shapes with given conditions

Drawing Geometric Shapes: Constructing Triangles and Exploring Uniqueness

Work through each section. For multiple-choice questions, select the best answer. For short-response questions, write complete sentences.

Grade 7·~35 min·Common Core Math - Grade 7·standard·7-g-a-2
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A

Recall / Warm-Up

1.

Which type of angle has a measure of 115°?

2.

In triangle XYZ, angle X measures 72° and angle Y measures 41°. What is the measure of angle Z?

3.

In triangle PQR, which angle is the included angle between sides PQ and PR?

B

Fluency Practice

1.

Which set of side lengths can form a valid triangle?

2.

A student constructs a triangle with sides 5 cm, 8 cm, and 6 cm using SSS. How many distinct triangles can be made with these three side lengths?

3.

Which condition always produces exactly one distinct triangle (up to congruence)?

4.

In triangle DEF, angle D measures 55° and angle E measures 48°. What is the measure of angle F, in degrees?

5.

You draw a triangle with angles 40°, 60°, and 80°. How many different triangles exist with these same three angle measures?

C

Varied Practice

1.

Which set of side lengths CANNOT form a triangle?

Triangle ABC with vertex A at left. Side AB = 6 cm runs horizontally right to B. Side AC = 4 cm runs up-right at 50° from AB to C. The 50° angle at A is marked with an arc. Both sides meeting at A are labeled.
2.

The figure shows triangle ABC with AB = 6 cm, AC = 4 cm, and angle A = 50°. Yuki says this is an SAS construction. Is Yuki correct, and how many distinct triangles have these measurements?

Three SSA construction panels side by side: Panel 1 shows a short arc that does not reach the far ray; Panel 2 shows an arc tangent to the far ray at one point; Panel 3 shows an arc from B with radius 7 cm (b = 10 cm, angle A = 35°) near the far ray — intersections not labeled.
3.

The figure shows an SSA setup: angle A = 35°, side b = 10 cm (from A to B), and side a = 7 cm (the arc radius from B). How many distinct triangles can be formed?

4.

Two students both draw a triangle using angles 45°, 65°, and 70°. Student 1 starts with a 4 cm base; Student 2 starts with an 8 cm base. Explain in one sentence why these two triangles are NOT identical, even though both satisfy the given angle conditions.

D

Word Problems

1.

Priya applies the triangle inequality before purchasing materials.

Priya wants to enclose a triangular garden using fence sections of lengths 5 m, 8 m, and 14 m. Without any tools, can she determine if the triangular garden is possible?

SSA construction with angle A = 40°, base AB = 10 cm, and a 7 cm arc centered at B sweeping upward toward the far ray of angle A. Intersection points are not marked.
2.

Kemal is designing a triangular support for a tent. Angle A (at one corner) measures 40°. The side from A to B is 10 cm, and the support pole connecting B to the far ray of angle A is 7 cm long. The diagram shows the arc of possible positions for the pole's far endpoint.

1.

How many distinct triangles can Kemal form using these measurements?

2.

One triangle has an acute angle at the top (at C₁). In that triangle, angle A = 40° and the top angle at C₁ measures 55°. What is the measure of angle B in that triangle?

A triangular roof truss with a 6 m horizontal base. The left base angle at L is 42° and the right base angle at R is 53°. The two sloped sides meet at peak P.
3.

Two builders fix a roof truss base at 6 m, then measure the angle at the left end as 42° and the angle at the right end as 53°. The figure shows the setup. How many distinct trusses can they build with these measurements?

4.

A tile designer makes triangular ceramic tiles with angles 50°, 60°, and 70°. She says all tiles with these three angles will be identical. Is she correct?

E

Error Analysis

Two-column contrast card: Maya's work (yellow) shows she stopped after finding one arc intersection and concluded 1 triangle, marked with a red X. Correct method (teal) shows drawing the full arc and counting both intersections, finding 2 triangles, with a check mark.
1.

Maya is given: angle A = 45°, side b = 8 cm (from A to B), and side a = 6 cm (the swinging side from B).

Maya's work:

  1. She draws angle A = 45° and marks B on the base ray at 8 cm from A.
  2. She sets her compass to 6 cm at B and draws an arc.
  3. The arc intersects the far ray of angle A at one point. She labels it C and draws triangle ABC.
  4. Maya writes: "I found one valid triangle. There is exactly one triangle with these measurements. Done."

Maya's conclusion is wrong. What error did she make?

What error did Maya make in her SSA triangle construction?

Two-column contrast card: Amir's claim (yellow) that three angle conditions give a unique triangle, marked wrong. Correct reasoning (teal) shows that the same angles with different base lengths (3 cm and 8 cm) both produce valid triangles, so AAA gives infinitely many similar triangles.
2.

Amir draws a triangle with angles 40°, 60°, and 80° and writes:

"I verified that 40° + 60° + 80° = 180°, so the triangle is valid. I gave three conditions, and my triangle is the only one satisfying all three. Therefore, AAA gives a unique triangle."

What is wrong with Amir's reasoning?

What error did Amir make in his reasoning about AAA triangle uniqueness?

F

Challenge

SSA ambiguous case diagram: angle A = 35°, AB = 10 cm. Arc from B with radius 7 cm crosses the far ray at two points C₁ and C₂. Triangle ABC₁ (solid lines) and triangle ABC₂ (dashed lines) share side AB and both have BC = 7 cm.
1.

The figure shows two triangles, ABC₁ and ABC₂, both satisfying: angle A = 35°, AB = 10 cm, and BC = 7 cm. Which statement is TRUE about these two triangles?

2.

Explain in 2–3 sentences why 'two sides and a non-included angle' (SSA) cannot serve as a triangle congruence condition. Include a specific numerical example to support your explanation.

0 of 21 answered