Practice · Chain isosceles base angles to find unknown angles · Sensim Math

Variant 1 easy

In the figure, sides $AB$ , $AC$ , and $CD$ have equal length, and the angle at point $D$ (angle $ADC$ ) measures $15\degree$ . Find the measure of angle $BAC$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Vertex A is at the top with B, C, D along the base. Sides AB, AC, and CD are all equal, and angle ADC is 15 degrees. I must find angle BAC.

Givens

AB = AC = CD (three equal segments)
B, C, D lie on the base in that order, so B, C, D are collinear
Angle ADC = 15 degrees

Unknowns

The measure of angle BAC

Constraints

An isosceles triangle has two equal base angles
Angles in a triangle sum to 180 degrees
Angles on a straight line sum to 180 degrees

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#11 Work Backwards

The figure splits into two isosceles triangles (ACD and ABC). I solve the one I can (ACD, where the known angle lives) first, carry the result across the straight base, then finish in triangle ABC.

3 · Execute3 carry out the plan

1Find angle ACD in isosceles triangle ACD

#7 Identify Subproblems 4.MD.C.7

Since AC = CD, triangle ACD is isosceles with equal base angles at A and D, so angle DAC = angle ADC = 15 degrees. Then angle ACD = 180 - 15 - 15 = 150 degrees.

\angle ACD = 180^\circ - 15^\circ - 15^\circ = 150^\circ

Equal sides give equal base angles, then the triangle-sum gives the third angle.

2Find angle ACB using the straight base

#7 Identify Subproblems 4.MD.C.7

B, C, D lie on one straight line, so angle ACB and angle ACD are a straight-line pair: angle ACB = 180 - 150 = 30 degrees.

\angle ACB = 180^\circ - 150^\circ = 30^\circ

Two angles that sit on a straight line always add to 180 degrees.

3Find angle BAC in isosceles triangle ABC

#7 Identify Subproblems 4.MD.C.7

Since AB = AC, triangle ABC is isosceles with equal base angles at B and C, so angle ABC = angle ACB = 30 degrees. Then angle BAC = 180 - 30 - 30 = 120 degrees.

\angle BAC = 180^\circ - 30^\circ - 30^\circ = 120^\circ

The same equal-base-angle rule, applied to the second isosceles triangle, finishes the chain.

Answer: 120 degrees

4 · Reviewdoes it hold up?

Angle BAC = 120 degrees, and each triangle's angles total 180 degrees (15+15+150 and 30+30+120), confirming the chain is consistent.

Another way: Draw the figure to scale (tool 1) and measure angle BAC with a protractor to confirm the 120-degree answer found by stepwise angle chasing.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Chaining isosceles base angles, the triangle-sum, and the straight-line angle across two triangles to find angle BAC.

💡Takeaway. This only needs Grade 4 angle-adding and the equal-base-angle rule you already know!

Variant 2 easy

In the figure, sides $AB$ , $AC$ , and $CD$ have equal length, and the angle at point $D$ (angle $ADC$ ) measures $20\degree$ . Find the measure of angle $BAC$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Vertex A is at the top with B, C, D along the base. Sides AB, AC, and CD are all equal, and angle ADC is 20 degrees. I must find angle BAC.

Givens

AB = AC = CD (three equal segments)
B, C, D lie on the base in that order, so B, C, D are collinear
Angle ADC = 20 degrees

Unknowns

The measure of angle BAC

Constraints

An isosceles triangle has two equal base angles
Angles in a triangle sum to 180 degrees
Angles on a straight line sum to 180 degrees

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#11 Work Backwards

The figure splits into two isosceles triangles (ACD and ABC). I solve the one I can (ACD, where the known angle lives) first, carry the result across the straight base, then finish in triangle ABC.

3 · Execute3 carry out the plan

1Find angle ACD in isosceles triangle ACD

#7 Identify Subproblems 4.MD.C.7

Since AC = CD, triangle ACD is isosceles with equal base angles at A and D, so angle DAC = angle ADC = 20 degrees. Then angle ACD = 180 - 20 - 20 = 140 degrees.

\angle ACD = 180^\circ - 20^\circ - 20^\circ = 140^\circ

Equal sides give equal base angles, then the triangle-sum gives the third angle.

2Find angle ACB using the straight base

#7 Identify Subproblems 4.MD.C.7

B, C, D lie on one straight line, so angle ACB and angle ACD are a straight-line pair: angle ACB = 180 - 140 = 40 degrees.

\angle ACB = 180^\circ - 140^\circ = 40^\circ

Two angles that sit on a straight line always add to 180 degrees.

3Find angle BAC in isosceles triangle ABC

#7 Identify Subproblems 4.MD.C.7

Since AB = AC, triangle ABC is isosceles with equal base angles at B and C, so angle ABC = angle ACB = 40 degrees. Then angle BAC = 180 - 40 - 40 = 100 degrees.

\angle BAC = 180^\circ - 40^\circ - 40^\circ = 100^\circ

The same equal-base-angle rule, applied to the second isosceles triangle, finishes the chain.

Answer: 100 degrees

4 · Reviewdoes it hold up?

Angle BAC = 100 degrees, and each triangle's angles total 180 degrees (20+20+140 and 40+40+100), confirming the chain is consistent.

Another way: Draw the figure to scale (tool 1) and measure angle BAC with a protractor to confirm the 100-degree answer found by stepwise angle chasing.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Chaining isosceles base angles, the triangle-sum, and the straight-line angle across two triangles to find angle BAC.

💡Takeaway. This only needs Grade 4 angle-adding and the equal-base-angle rule you already know!

Variant 3 easy

In the figure, sides $AB$ , $AC$ , and $CD$ have equal length, and the angle at point $D$ (angle $ADC$ ) measures $22\degree$ . Find the measure of angle $BAC$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Vertex A is at the top with B, C, D along the base. Sides AB, AC, and CD are all equal, and angle ADC is 22 degrees. I must find angle BAC.

Givens

AB = AC = CD (three equal segments)
B, C, D lie on the base in that order, so B, C, D are collinear
Angle ADC = 22 degrees

Unknowns

The measure of angle BAC

Constraints

An isosceles triangle has two equal base angles
Angles in a triangle sum to 180 degrees
Angles on a straight line sum to 180 degrees

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#11 Work Backwards

The figure splits into two isosceles triangles (ACD and ABC). I solve the one I can (ACD, where the known angle lives) first, carry the result across the straight base, then finish in triangle ABC.

3 · Execute3 carry out the plan

1Find angle ACD in isosceles triangle ACD

#7 Identify Subproblems 4.MD.C.7

Since AC = CD, triangle ACD is isosceles with equal base angles at A and D, so angle DAC = angle ADC = 22 degrees. Then angle ACD = 180 - 22 - 22 = 136 degrees.

\angle ACD = 180^\circ - 22^\circ - 22^\circ = 136^\circ

Equal sides give equal base angles, then the triangle-sum gives the third angle.

2Find angle ACB using the straight base

#7 Identify Subproblems 4.MD.C.7

B, C, D lie on one straight line, so angle ACB and angle ACD are a straight-line pair: angle ACB = 180 - 136 = 44 degrees.

\angle ACB = 180^\circ - 136^\circ = 44^\circ

Two angles that sit on a straight line always add to 180 degrees.

3Find angle BAC in isosceles triangle ABC

#7 Identify Subproblems 4.MD.C.7

Since AB = AC, triangle ABC is isosceles with equal base angles at B and C, so angle ABC = angle ACB = 44 degrees. Then angle BAC = 180 - 44 - 44 = 92 degrees.

\angle BAC = 180^\circ - 44^\circ - 44^\circ = 92^\circ

The same equal-base-angle rule, applied to the second isosceles triangle, finishes the chain.

Answer: 92 degrees

4 · Reviewdoes it hold up?

Angle BAC = 92 degrees, and each triangle's angles total 180 degrees (22+22+136 and 44+44+92), confirming the chain is consistent.

Another way: Draw the figure to scale (tool 1) and measure angle BAC with a protractor to confirm the 92-degree answer found by stepwise angle chasing.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Chaining isosceles base angles, the triangle-sum, and the straight-line angle across two triangles to find angle BAC.

💡Takeaway. This only needs Grade 4 angle-adding and the equal-base-angle rule you already know!

Variant 4 medium

In the figure, sides $AB$ , $AC$ , and $CD$ have equal length, and the angle at point $D$ (angle $ADC$ ) measures $25\degree$ . Find the measure of angle $BAC$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Vertex A is at the top with B, C, D along the base. Sides AB, AC, and CD are all equal, and angle ADC is 25 degrees. I must find angle BAC.

Givens

AB = AC = CD (three equal segments)
B, C, D lie on the base in that order, so B, C, D are collinear
Angle ADC = 25 degrees

Unknowns

The measure of angle BAC

Constraints

An isosceles triangle has two equal base angles
Angles in a triangle sum to 180 degrees
Angles on a straight line sum to 180 degrees

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#11 Work Backwards

The figure splits into two isosceles triangles (ACD and ABC). I solve the one I can (ACD, where the known angle lives) first, carry the result across the straight base, then finish in triangle ABC.

3 · Execute3 carry out the plan

1Find angle ACD in isosceles triangle ACD

#7 Identify Subproblems 4.MD.C.7

Since AC = CD, triangle ACD is isosceles with equal base angles at A and D, so angle DAC = angle ADC = 25 degrees. Then angle ACD = 180 - 25 - 25 = 130 degrees.

\angle ACD = 180^\circ - 25^\circ - 25^\circ = 130^\circ

Equal sides give equal base angles, then the triangle-sum gives the third angle.

2Find angle ACB using the straight base

#7 Identify Subproblems 4.MD.C.7

B, C, D lie on one straight line, so angle ACB and angle ACD are a straight-line pair: angle ACB = 180 - 130 = 50 degrees.

\angle ACB = 180^\circ - 130^\circ = 50^\circ

Two angles that sit on a straight line always add to 180 degrees.

3Find angle BAC in isosceles triangle ABC

#7 Identify Subproblems 4.MD.C.7

Since AB = AC, triangle ABC is isosceles with equal base angles at B and C, so angle ABC = angle ACB = 50 degrees. Then angle BAC = 180 - 50 - 50 = 80 degrees.

\angle BAC = 180^\circ - 50^\circ - 50^\circ = 80^\circ

The same equal-base-angle rule, applied to the second isosceles triangle, finishes the chain.

Answer: 80 degrees

4 · Reviewdoes it hold up?

Angle BAC = 80 degrees, and each triangle's angles total 180 degrees (25+25+130 and 50+50+80), confirming the chain is consistent.

Another way: Draw the figure to scale (tool 1) and measure angle BAC with a protractor to confirm the 80-degree answer found by stepwise angle chasing.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Chaining isosceles base angles, the triangle-sum, and the straight-line angle across two triangles to find angle BAC.

💡Takeaway. This only needs Grade 4 angle-adding and the equal-base-angle rule you already know!

Variant 5 medium

In the figure, sides $AB$ , $AC$ , and $CD$ have equal length, and the angle at point $D$ (angle $ADC$ ) measures $28\degree$ . Find the measure of angle $BAC$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Vertex A is at the top with B, C, D along the base. Sides AB, AC, and CD are all equal, and angle ADC is 28 degrees. I must find angle BAC.

Givens

AB = AC = CD (three equal segments)
B, C, D lie on the base in that order, so B, C, D are collinear
Angle ADC = 28 degrees

Unknowns

The measure of angle BAC

Constraints

An isosceles triangle has two equal base angles
Angles in a triangle sum to 180 degrees
Angles on a straight line sum to 180 degrees

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#11 Work Backwards

The figure splits into two isosceles triangles (ACD and ABC). I solve the one I can (ACD, where the known angle lives) first, carry the result across the straight base, then finish in triangle ABC.

3 · Execute3 carry out the plan

1Find angle ACD in isosceles triangle ACD

#7 Identify Subproblems 4.MD.C.7

Since AC = CD, triangle ACD is isosceles with equal base angles at A and D, so angle DAC = angle ADC = 28 degrees. Then angle ACD = 180 - 28 - 28 = 124 degrees.

\angle ACD = 180^\circ - 28^\circ - 28^\circ = 124^\circ

Equal sides give equal base angles, then the triangle-sum gives the third angle.

2Find angle ACB using the straight base

#7 Identify Subproblems 4.MD.C.7

B, C, D lie on one straight line, so angle ACB and angle ACD are a straight-line pair: angle ACB = 180 - 124 = 56 degrees.

\angle ACB = 180^\circ - 124^\circ = 56^\circ

Two angles that sit on a straight line always add to 180 degrees.

3Find angle BAC in isosceles triangle ABC

#7 Identify Subproblems 4.MD.C.7

Since AB = AC, triangle ABC is isosceles with equal base angles at B and C, so angle ABC = angle ACB = 56 degrees. Then angle BAC = 180 - 56 - 56 = 68 degrees.

\angle BAC = 180^\circ - 56^\circ - 56^\circ = 68^\circ

The same equal-base-angle rule, applied to the second isosceles triangle, finishes the chain.

Answer: 68 degrees

4 · Reviewdoes it hold up?

Angle BAC = 68 degrees, and each triangle's angles total 180 degrees (28+28+124 and 56+56+68), confirming the chain is consistent.

Another way: Draw the figure to scale (tool 1) and measure angle BAC with a protractor to confirm the 68-degree answer found by stepwise angle chasing.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Chaining isosceles base angles, the triangle-sum, and the straight-line angle across two triangles to find angle BAC.

💡Takeaway. This only needs Grade 4 angle-adding and the equal-base-angle rule you already know!

Variant 6 medium

In the figure, sides $AB$ , $AC$ , and $CD$ have equal length, and the angle at point $D$ (angle $ADC$ ) measures $30\degree$ . Find the measure of angle $BAC$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Vertex A is at the top with B, C, D along the base. Sides AB, AC, and CD are all equal, and angle ADC is 30 degrees. I must find angle BAC.

Givens

AB = AC = CD (three equal segments)
B, C, D lie on the base in that order, so B, C, D are collinear
Angle ADC = 30 degrees

Unknowns

The measure of angle BAC

Constraints

An isosceles triangle has two equal base angles
Angles in a triangle sum to 180 degrees
Angles on a straight line sum to 180 degrees

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#11 Work Backwards

The figure splits into two isosceles triangles (ACD and ABC). I solve the one I can (ACD, where the known angle lives) first, carry the result across the straight base, then finish in triangle ABC.

3 · Execute3 carry out the plan

1Find angle ACD in isosceles triangle ACD

#7 Identify Subproblems 4.MD.C.7

Since AC = CD, triangle ACD is isosceles with equal base angles at A and D, so angle DAC = angle ADC = 30 degrees. Then angle ACD = 180 - 30 - 30 = 120 degrees.

\angle ACD = 180^\circ - 30^\circ - 30^\circ = 120^\circ

Equal sides give equal base angles, then the triangle-sum gives the third angle.

2Find angle ACB using the straight base

#7 Identify Subproblems 4.MD.C.7

B, C, D lie on one straight line, so angle ACB and angle ACD are a straight-line pair: angle ACB = 180 - 120 = 60 degrees.

\angle ACB = 180^\circ - 120^\circ = 60^\circ

Two angles that sit on a straight line always add to 180 degrees.

3Find angle BAC in isosceles triangle ABC

#7 Identify Subproblems 4.MD.C.7

Since AB = AC, triangle ABC is isosceles with equal base angles at B and C, so angle ABC = angle ACB = 60 degrees. Then angle BAC = 180 - 60 - 60 = 60 degrees.

\angle BAC = 180^\circ - 60^\circ - 60^\circ = 60^\circ

The same equal-base-angle rule, applied to the second isosceles triangle, finishes the chain.

Answer: 60 degrees

4 · Reviewdoes it hold up?

Angle BAC = 60 degrees, and each triangle's angles total 180 degrees (30+30+120 and 60+60+60), confirming the chain is consistent.

Another way: Draw the figure to scale (tool 1) and measure angle BAC with a protractor to confirm the 60-degree answer found by stepwise angle chasing.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Chaining isosceles base angles, the triangle-sum, and the straight-line angle across two triangles to find angle BAC.

💡Takeaway. This only needs Grade 4 angle-adding and the equal-base-angle rule you already know!

Variant 7 medium

In the figure, sides $AB$ , $AC$ , and $CD$ have equal length, and the angle at point $D$ (angle $ADC$ ) measures $32\degree$ . Find the measure of angle $BAC$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Vertex A is at the top with B, C, D along the base. Sides AB, AC, and CD are all equal, and angle ADC is 32 degrees. I must find angle BAC.

Givens

AB = AC = CD (three equal segments)
B, C, D lie on the base in that order, so B, C, D are collinear
Angle ADC = 32 degrees

Unknowns

The measure of angle BAC

Constraints

An isosceles triangle has two equal base angles
Angles in a triangle sum to 180 degrees
Angles on a straight line sum to 180 degrees

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#11 Work Backwards

The figure splits into two isosceles triangles (ACD and ABC). I solve the one I can (ACD, where the known angle lives) first, carry the result across the straight base, then finish in triangle ABC.

3 · Execute3 carry out the plan

1Find angle ACD in isosceles triangle ACD

#7 Identify Subproblems 4.MD.C.7

Since AC = CD, triangle ACD is isosceles with equal base angles at A and D, so angle DAC = angle ADC = 32 degrees. Then angle ACD = 180 - 32 - 32 = 116 degrees.

\angle ACD = 180^\circ - 32^\circ - 32^\circ = 116^\circ

Equal sides give equal base angles, then the triangle-sum gives the third angle.

2Find angle ACB using the straight base

#7 Identify Subproblems 4.MD.C.7

B, C, D lie on one straight line, so angle ACB and angle ACD are a straight-line pair: angle ACB = 180 - 116 = 64 degrees.

\angle ACB = 180^\circ - 116^\circ = 64^\circ

Two angles that sit on a straight line always add to 180 degrees.

3Find angle BAC in isosceles triangle ABC

#7 Identify Subproblems 4.MD.C.7

Since AB = AC, triangle ABC is isosceles with equal base angles at B and C, so angle ABC = angle ACB = 64 degrees. Then angle BAC = 180 - 64 - 64 = 52 degrees.

\angle BAC = 180^\circ - 64^\circ - 64^\circ = 52^\circ

The same equal-base-angle rule, applied to the second isosceles triangle, finishes the chain.

Answer: 52 degrees

4 · Reviewdoes it hold up?

Angle BAC = 52 degrees, and each triangle's angles total 180 degrees (32+32+116 and 64+64+52), confirming the chain is consistent.

Another way: Draw the figure to scale (tool 1) and measure angle BAC with a protractor to confirm the 52-degree answer found by stepwise angle chasing.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Chaining isosceles base angles, the triangle-sum, and the straight-line angle across two triangles to find angle BAC.

💡Takeaway. This only needs Grade 4 angle-adding and the equal-base-angle rule you already know!

Variant 8 hard

In the figure, sides $AB$ , $AC$ , and $CD$ have equal length, and the angle at point $D$ (angle $ADC$ ) measures $35\degree$ . Find the measure of angle $BAC$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Vertex A is at the top with B, C, D along the base. Sides AB, AC, and CD are all equal, and angle ADC is 35 degrees. I must find angle BAC.

Givens

AB = AC = CD (three equal segments)
B, C, D lie on the base in that order, so B, C, D are collinear
Angle ADC = 35 degrees

Unknowns

The measure of angle BAC

Constraints

An isosceles triangle has two equal base angles
Angles in a triangle sum to 180 degrees
Angles on a straight line sum to 180 degrees

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#11 Work Backwards

The figure splits into two isosceles triangles (ACD and ABC). I solve the one I can (ACD, where the known angle lives) first, carry the result across the straight base, then finish in triangle ABC.

3 · Execute3 carry out the plan

1Find angle ACD in isosceles triangle ACD

#7 Identify Subproblems 4.MD.C.7

Since AC = CD, triangle ACD is isosceles with equal base angles at A and D, so angle DAC = angle ADC = 35 degrees. Then angle ACD = 180 - 35 - 35 = 110 degrees.

\angle ACD = 180^\circ - 35^\circ - 35^\circ = 110^\circ

Equal sides give equal base angles, then the triangle-sum gives the third angle.

2Find angle ACB using the straight base

#7 Identify Subproblems 4.MD.C.7

B, C, D lie on one straight line, so angle ACB and angle ACD are a straight-line pair: angle ACB = 180 - 110 = 70 degrees.

\angle ACB = 180^\circ - 110^\circ = 70^\circ

Two angles that sit on a straight line always add to 180 degrees.

3Find angle BAC in isosceles triangle ABC

#7 Identify Subproblems 4.MD.C.7

Since AB = AC, triangle ABC is isosceles with equal base angles at B and C, so angle ABC = angle ACB = 70 degrees. Then angle BAC = 180 - 70 - 70 = 40 degrees.

\angle BAC = 180^\circ - 70^\circ - 70^\circ = 40^\circ

The same equal-base-angle rule, applied to the second isosceles triangle, finishes the chain.

Answer: 40 degrees

4 · Reviewdoes it hold up?

Angle BAC = 40 degrees, and each triangle's angles total 180 degrees (35+35+110 and 70+70+40), confirming the chain is consistent.

Another way: Draw the figure to scale (tool 1) and measure angle BAC with a protractor to confirm the 40-degree answer found by stepwise angle chasing.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Chaining isosceles base angles, the triangle-sum, and the straight-line angle across two triangles to find angle BAC.

💡Takeaway. This only needs Grade 4 angle-adding and the equal-base-angle rule you already know!

Variant 9 hard

In the figure, sides $AB$ , $AC$ , and $CD$ have equal length, and the angle at point $D$ (angle $ADC$ ) measures $38\degree$ . Find the measure of angle $BAC$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Vertex A is at the top with B, C, D along the base. Sides AB, AC, and CD are all equal, and angle ADC is 38 degrees. I must find angle BAC.

Givens

AB = AC = CD (three equal segments)
B, C, D lie on the base in that order, so B, C, D are collinear
Angle ADC = 38 degrees

Unknowns

The measure of angle BAC

Constraints

An isosceles triangle has two equal base angles
Angles in a triangle sum to 180 degrees
Angles on a straight line sum to 180 degrees

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#11 Work Backwards

The figure splits into two isosceles triangles (ACD and ABC). I solve the one I can (ACD, where the known angle lives) first, carry the result across the straight base, then finish in triangle ABC.

3 · Execute3 carry out the plan

1Find angle ACD in isosceles triangle ACD

#7 Identify Subproblems 4.MD.C.7

Since AC = CD, triangle ACD is isosceles with equal base angles at A and D, so angle DAC = angle ADC = 38 degrees. Then angle ACD = 180 - 38 - 38 = 104 degrees.

\angle ACD = 180^\circ - 38^\circ - 38^\circ = 104^\circ

Equal sides give equal base angles, then the triangle-sum gives the third angle.

2Find angle ACB using the straight base

#7 Identify Subproblems 4.MD.C.7

B, C, D lie on one straight line, so angle ACB and angle ACD are a straight-line pair: angle ACB = 180 - 104 = 76 degrees.

\angle ACB = 180^\circ - 104^\circ = 76^\circ

Two angles that sit on a straight line always add to 180 degrees.

3Find angle BAC in isosceles triangle ABC

#7 Identify Subproblems 4.MD.C.7

Since AB = AC, triangle ABC is isosceles with equal base angles at B and C, so angle ABC = angle ACB = 76 degrees. Then angle BAC = 180 - 76 - 76 = 28 degrees.

\angle BAC = 180^\circ - 76^\circ - 76^\circ = 28^\circ

The same equal-base-angle rule, applied to the second isosceles triangle, finishes the chain.

Answer: 28 degrees

4 · Reviewdoes it hold up?

Angle BAC = 28 degrees, and each triangle's angles total 180 degrees (38+38+104 and 76+76+28), confirming the chain is consistent.

Another way: Draw the figure to scale (tool 1) and measure angle BAC with a protractor to confirm the 28-degree answer found by stepwise angle chasing.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Chaining isosceles base angles, the triangle-sum, and the straight-line angle across two triangles to find angle BAC.

💡Takeaway. This only needs Grade 4 angle-adding and the equal-base-angle rule you already know!

Variant 10 hard

In the figure, sides $AB$ , $AC$ , and $CD$ have equal length, and the angle at point $D$ (angle $ADC$ ) measures $40\degree$ . Find the measure of angle $BAC$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Vertex A is at the top with B, C, D along the base. Sides AB, AC, and CD are all equal, and angle ADC is 40 degrees. I must find angle BAC.

Givens

AB = AC = CD (three equal segments)
B, C, D lie on the base in that order, so B, C, D are collinear
Angle ADC = 40 degrees

Unknowns

The measure of angle BAC

Constraints

An isosceles triangle has two equal base angles
Angles in a triangle sum to 180 degrees
Angles on a straight line sum to 180 degrees

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#11 Work Backwards

The figure splits into two isosceles triangles (ACD and ABC). I solve the one I can (ACD, where the known angle lives) first, carry the result across the straight base, then finish in triangle ABC.

3 · Execute3 carry out the plan

1Find angle ACD in isosceles triangle ACD

#7 Identify Subproblems 4.MD.C.7

Since AC = CD, triangle ACD is isosceles with equal base angles at A and D, so angle DAC = angle ADC = 40 degrees. Then angle ACD = 180 - 40 - 40 = 100 degrees.

\angle ACD = 180^\circ - 40^\circ - 40^\circ = 100^\circ

Equal sides give equal base angles, then the triangle-sum gives the third angle.

2Find angle ACB using the straight base

#7 Identify Subproblems 4.MD.C.7

B, C, D lie on one straight line, so angle ACB and angle ACD are a straight-line pair: angle ACB = 180 - 100 = 80 degrees.

\angle ACB = 180^\circ - 100^\circ = 80^\circ

Two angles that sit on a straight line always add to 180 degrees.

3Find angle BAC in isosceles triangle ABC

#7 Identify Subproblems 4.MD.C.7

Since AB = AC, triangle ABC is isosceles with equal base angles at B and C, so angle ABC = angle ACB = 80 degrees. Then angle BAC = 180 - 80 - 80 = 20 degrees.

\angle BAC = 180^\circ - 80^\circ - 80^\circ = 20^\circ

The same equal-base-angle rule, applied to the second isosceles triangle, finishes the chain.

Answer: 20 degrees

4 · Reviewdoes it hold up?

Angle BAC = 20 degrees, and each triangle's angles total 180 degrees (40+40+100 and 80+80+20), confirming the chain is consistent.

Another way: Draw the figure to scale (tool 1) and measure angle BAC with a protractor to confirm the 20-degree answer found by stepwise angle chasing.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Chaining isosceles base angles, the triangle-sum, and the straight-line angle across two triangles to find angle BAC.

💡Takeaway. This only needs Grade 4 angle-adding and the equal-base-angle rule you already know!

Chain isosceles base angles to find unknown angles · 10 practice problems

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