Practice · Set up an equation with one unknown angle · Sensim Math

Variant 1 medium

Angle $\angle BOC$ is $10^\circ$ larger than angle $\angle COD$ . Using the figure, find the measure of angle $\angle COD$ .

[Figure] Straight line $AD$ passes through point $O$ , and two rays $OB$ and $OC$ extend upward from $O$ . Starting from the left end of the line, the angles $\angle AOB = 50^\circ$ , $\angle BOC$ , and $\angle COD$ lie in order, so the three angles together form the straight angle $180^\circ$ .

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1 · Understandwhat's really being asked

On straight line AD through point O, rays OB and OC go upward. From the left, the angles AOB = 50 degrees, BOC, and COD line up and together make the straight angle of 180 degrees. Also angle BOC is 10 degrees bigger than angle COD. Find angle COD.

Givens

Angle AOB = 50 degrees.
Angles AOB, BOC, COD sit in order along straight line AD, so they add to 180 degrees.
Angle BOC = angle COD + 10 degrees.

Unknowns

The measure of angle COD.

Constraints

Angles on a straight line add to 180 degrees.
BOC is exactly 10 degrees larger than COD.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

First peel off the known 50 degrees to find how much BOC and COD share. Then, since BOC is 10 more than COD, take that extra off the shared total and split the rest into the two equal parts; guess-and-check confirms COD.

3 · Execute2 carry out the plan

1Find the total for BOC and COD

#7 Identify Subproblems 4.MD.C.7

The three angles fill the 180-degree straight line, so BOC plus COD is the part left after the known angle.

\angle BOC + \angle COD = 180^\circ - 50^\circ = 130^\circ

A flat line is 180 degrees; remove the known piece and the other two pieces share the rest.

2Remove the extra 10 degrees and split

#6 Guess and Check 4.MD.C.7

BOC is COD plus 10. If we set aside that extra 10 degrees from the 130, the leftover 120 degrees splits into two equal COD-sized parts. Half of 120 is the measure of COD.

(130^\circ - 10^\circ) \div 2 = 120^\circ \div 2 = 60^\circ

Two parts differ by a fixed amount; take the extra away so they match, share equally, and that equal part is the smaller angle.

Answer: 60 degrees

4 · Reviewdoes it hold up?

If COD = 60 degrees, then BOC = 70 degrees. Check the line: 50 + 70 + 60 = 180 degrees, and 70 is exactly 10 more than 60. Everything fits.

Another way: Convert to algebra (tool 13): let COD = x, then BOC = x + 10, and 50 + (x + 10) + x = 180, giving 2x = 120, x = 60 degrees.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight-line total and the degree difference to find COD.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the rays OB, OC and the line AD through point O.

💡Takeaway. Take the known angle off the 180-degree line, set aside the extra, then share what is left in two - all Grade 4 angle adding and subtracting!

Variant 2 medium

Angle $\angle BOC$ is $20^\circ$ larger than angle $\angle COD$ . Using the figure, find the measure of angle $\angle COD$ .

[Figure] Straight line $AD$ passes through point $O$ , and two rays $OB$ and $OC$ extend upward from $O$ . Starting from the left end of the line, the angles $\angle AOB = 60^\circ$ , $\angle BOC$ , and $\angle COD$ lie in order, so the three angles together form the straight angle $180^\circ$ .

Your answer

Show solution

1 · Understandwhat's really being asked

On straight line AD through point O, rays OB and OC go upward. From the left, the angles AOB = 60 degrees, BOC, and COD line up and together make the straight angle of 180 degrees. Also angle BOC is 20 degrees bigger than angle COD. Find angle COD.

Givens

Angle AOB = 60 degrees.
Angles AOB, BOC, COD sit in order along straight line AD, so they add to 180 degrees.
Angle BOC = angle COD + 20 degrees.

Unknowns

The measure of angle COD.

Constraints

Angles on a straight line add to 180 degrees.
BOC is exactly 20 degrees larger than COD.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

First peel off the known 60 degrees to find how much BOC and COD share. Then, since BOC is 20 more than COD, take that extra off the shared total and split the rest into the two equal parts; guess-and-check confirms COD.

3 · Execute2 carry out the plan

1Find the total for BOC and COD

#7 Identify Subproblems 4.MD.C.7

The three angles fill the 180-degree straight line, so BOC plus COD is the part left after the known angle.

\angle BOC + \angle COD = 180^\circ - 60^\circ = 120^\circ

A flat line is 180 degrees; remove the known piece and the other two pieces share the rest.

2Remove the extra 20 degrees and split

#6 Guess and Check 4.MD.C.7

BOC is COD plus 20. If we set aside that extra 20 degrees from the 120, the leftover 100 degrees splits into two equal COD-sized parts. Half of 100 is the measure of COD.

(120^\circ - 20^\circ) \div 2 = 100^\circ \div 2 = 50^\circ

Two parts differ by a fixed amount; take the extra away so they match, share equally, and that equal part is the smaller angle.

Answer: 50 degrees

4 · Reviewdoes it hold up?

If COD = 50 degrees, then BOC = 70 degrees. Check the line: 60 + 70 + 50 = 180 degrees, and 70 is exactly 20 more than 50. Everything fits.

Another way: Convert to algebra (tool 13): let COD = x, then BOC = x + 20, and 60 + (x + 20) + x = 180, giving 2x = 100, x = 50 degrees.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight-line total and the degree difference to find COD.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the rays OB, OC and the line AD through point O.

💡Takeaway. Take the known angle off the 180-degree line, set aside the extra, then share what is left in two - all Grade 4 angle adding and subtracting!

Variant 3 medium

Angle $\angle BOC$ is $40^\circ$ larger than angle $\angle COD$ . Using the figure, find the measure of angle $\angle COD$ .

[Figure] Straight line $AD$ passes through point $O$ , and two rays $OB$ and $OC$ extend upward from $O$ . Starting from the left end of the line, the angles $\angle AOB = 100^\circ$ , $\angle BOC$ , and $\angle COD$ lie in order, so the three angles together form the straight angle $180^\circ$ .

Your answer

Show solution

1 · Understandwhat's really being asked

On straight line AD through point O, rays OB and OC go upward. From the left, the angles AOB = 100 degrees, BOC, and COD line up and together make the straight angle of 180 degrees. Also angle BOC is 40 degrees bigger than angle COD. Find angle COD.

Givens

Angle AOB = 100 degrees.
Angles AOB, BOC, COD sit in order along straight line AD, so they add to 180 degrees.
Angle BOC = angle COD + 40 degrees.

Unknowns

The measure of angle COD.

Constraints

Angles on a straight line add to 180 degrees.
BOC is exactly 40 degrees larger than COD.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

First peel off the known 100 degrees to find how much BOC and COD share. Then, since BOC is 40 more than COD, take that extra off the shared total and split the rest into the two equal parts; guess-and-check confirms COD.

3 · Execute2 carry out the plan

1Find the total for BOC and COD

#7 Identify Subproblems 4.MD.C.7

The three angles fill the 180-degree straight line, so BOC plus COD is the part left after the known angle.

\angle BOC + \angle COD = 180^\circ - 100^\circ = 80^\circ

A flat line is 180 degrees; remove the known piece and the other two pieces share the rest.

2Remove the extra 40 degrees and split

#6 Guess and Check 4.MD.C.7

BOC is COD plus 40. If we set aside that extra 40 degrees from the 80, the leftover 40 degrees splits into two equal COD-sized parts. Half of 40 is the measure of COD.

(80^\circ - 40^\circ) \div 2 = 40^\circ \div 2 = 20^\circ

Two parts differ by a fixed amount; take the extra away so they match, share equally, and that equal part is the smaller angle.

Answer: 20 degrees

4 · Reviewdoes it hold up?

If COD = 20 degrees, then BOC = 60 degrees. Check the line: 100 + 60 + 20 = 180 degrees, and 60 is exactly 40 more than 20. Everything fits.

Another way: Convert to algebra (tool 13): let COD = x, then BOC = x + 40, and 100 + (x + 40) + x = 180, giving 2x = 40, x = 20 degrees.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight-line total and the degree difference to find COD.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the rays OB, OC and the line AD through point O.

💡Takeaway. Take the known angle off the 180-degree line, set aside the extra, then share what is left in two - all Grade 4 angle adding and subtracting!

Variant 4 medium

Angle $\angle BOC$ is $20^\circ$ larger than angle $\angle COD$ . Using the figure, find the measure of angle $\angle COD$ .

[Figure] Straight line $AD$ passes through point $O$ , and two rays $OB$ and $OC$ extend upward from $O$ . Starting from the left end of the line, the angles $\angle AOB = 80^\circ$ , $\angle BOC$ , and $\angle COD$ lie in order, so the three angles together form the straight angle $180^\circ$ .

Your answer

Show solution

1 · Understandwhat's really being asked

On straight line AD through point O, rays OB and OC go upward. From the left, the angles AOB = 80 degrees, BOC, and COD line up and together make the straight angle of 180 degrees. Also angle BOC is 20 degrees bigger than angle COD. Find angle COD.

Givens

Angle AOB = 80 degrees.
Angles AOB, BOC, COD sit in order along straight line AD, so they add to 180 degrees.
Angle BOC = angle COD + 20 degrees.

Unknowns

The measure of angle COD.

Constraints

Angles on a straight line add to 180 degrees.
BOC is exactly 20 degrees larger than COD.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

First peel off the known 80 degrees to find how much BOC and COD share. Then, since BOC is 20 more than COD, take that extra off the shared total and split the rest into the two equal parts; guess-and-check confirms COD.

3 · Execute2 carry out the plan

1Find the total for BOC and COD

#7 Identify Subproblems 4.MD.C.7

The three angles fill the 180-degree straight line, so BOC plus COD is the part left after the known angle.

\angle BOC + \angle COD = 180^\circ - 80^\circ = 100^\circ

A flat line is 180 degrees; remove the known piece and the other two pieces share the rest.

2Remove the extra 20 degrees and split

#6 Guess and Check 4.MD.C.7

BOC is COD plus 20. If we set aside that extra 20 degrees from the 100, the leftover 80 degrees splits into two equal COD-sized parts. Half of 80 is the measure of COD.

(100^\circ - 20^\circ) \div 2 = 80^\circ \div 2 = 40^\circ

Two parts differ by a fixed amount; take the extra away so they match, share equally, and that equal part is the smaller angle.

Answer: 40 degrees

4 · Reviewdoes it hold up?

If COD = 40 degrees, then BOC = 60 degrees. Check the line: 80 + 60 + 40 = 180 degrees, and 60 is exactly 20 more than 40. Everything fits.

Another way: Convert to algebra (tool 13): let COD = x, then BOC = x + 20, and 80 + (x + 20) + x = 180, giving 2x = 80, x = 40 degrees.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight-line total and the degree difference to find COD.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the rays OB, OC and the line AD through point O.

💡Takeaway. Take the known angle off the 180-degree line, set aside the extra, then share what is left in two - all Grade 4 angle adding and subtracting!

Variant 5 medium

Angle $\angle BOC$ is $60^\circ$ larger than angle $\angle COD$ . Using the figure, find the measure of angle $\angle COD$ .

[Figure] Straight line $AD$ passes through point $O$ , and two rays $OB$ and $OC$ extend upward from $O$ . Starting from the left end of the line, the angles $\angle AOB = 20^\circ$ , $\angle BOC$ , and $\angle COD$ lie in order, so the three angles together form the straight angle $180^\circ$ .

Your answer

Show solution

1 · Understandwhat's really being asked

On straight line AD through point O, rays OB and OC go upward. From the left, the angles AOB = 20 degrees, BOC, and COD line up and together make the straight angle of 180 degrees. Also angle BOC is 60 degrees bigger than angle COD. Find angle COD.

Givens

Angle AOB = 20 degrees.
Angles AOB, BOC, COD sit in order along straight line AD, so they add to 180 degrees.
Angle BOC = angle COD + 60 degrees.

Unknowns

The measure of angle COD.

Constraints

Angles on a straight line add to 180 degrees.
BOC is exactly 60 degrees larger than COD.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

First peel off the known 20 degrees to find how much BOC and COD share. Then, since BOC is 60 more than COD, take that extra off the shared total and split the rest into the two equal parts; guess-and-check confirms COD.

3 · Execute2 carry out the plan

1Find the total for BOC and COD

#7 Identify Subproblems 4.MD.C.7

The three angles fill the 180-degree straight line, so BOC plus COD is the part left after the known angle.

\angle BOC + \angle COD = 180^\circ - 20^\circ = 160^\circ

A flat line is 180 degrees; remove the known piece and the other two pieces share the rest.

2Remove the extra 60 degrees and split

#6 Guess and Check 4.MD.C.7

BOC is COD plus 60. If we set aside that extra 60 degrees from the 160, the leftover 100 degrees splits into two equal COD-sized parts. Half of 100 is the measure of COD.

(160^\circ - 60^\circ) \div 2 = 100^\circ \div 2 = 50^\circ

Two parts differ by a fixed amount; take the extra away so they match, share equally, and that equal part is the smaller angle.

Answer: 50 degrees

4 · Reviewdoes it hold up?

If COD = 50 degrees, then BOC = 110 degrees. Check the line: 20 + 110 + 50 = 180 degrees, and 110 is exactly 60 more than 50. Everything fits.

Another way: Convert to algebra (tool 13): let COD = x, then BOC = x + 60, and 20 + (x + 60) + x = 180, giving 2x = 100, x = 50 degrees.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight-line total and the degree difference to find COD.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the rays OB, OC and the line AD through point O.

💡Takeaway. Take the known angle off the 180-degree line, set aside the extra, then share what is left in two - all Grade 4 angle adding and subtracting!

Variant 6 medium

Angle $\angle BOC$ is $30^\circ$ larger than angle $\angle COD$ . Using the figure, find the measure of angle $\angle COD$ .

[Figure] Straight line $AD$ passes through point $O$ , and two rays $OB$ and $OC$ extend upward from $O$ . Starting from the left end of the line, the angles $\angle AOB = 40^\circ$ , $\angle BOC$ , and $\angle COD$ lie in order, so the three angles together form the straight angle $180^\circ$ .

Your answer

Show solution

1 · Understandwhat's really being asked

On straight line AD through point O, rays OB and OC go upward. From the left, the angles AOB = 40 degrees, BOC, and COD line up and together make the straight angle of 180 degrees. Also angle BOC is 30 degrees bigger than angle COD. Find angle COD.

Givens

Angle AOB = 40 degrees.
Angles AOB, BOC, COD sit in order along straight line AD, so they add to 180 degrees.
Angle BOC = angle COD + 30 degrees.

Unknowns

The measure of angle COD.

Constraints

Angles on a straight line add to 180 degrees.
BOC is exactly 30 degrees larger than COD.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

First peel off the known 40 degrees to find how much BOC and COD share. Then, since BOC is 30 more than COD, take that extra off the shared total and split the rest into the two equal parts; guess-and-check confirms COD.

3 · Execute2 carry out the plan

1Find the total for BOC and COD

#7 Identify Subproblems 4.MD.C.7

The three angles fill the 180-degree straight line, so BOC plus COD is the part left after the known angle.

\angle BOC + \angle COD = 180^\circ - 40^\circ = 140^\circ

A flat line is 180 degrees; remove the known piece and the other two pieces share the rest.

2Remove the extra 30 degrees and split

#6 Guess and Check 4.MD.C.7

BOC is COD plus 30. If we set aside that extra 30 degrees from the 140, the leftover 110 degrees splits into two equal COD-sized parts. Half of 110 is the measure of COD.

(140^\circ - 30^\circ) \div 2 = 110^\circ \div 2 = 55^\circ

Two parts differ by a fixed amount; take the extra away so they match, share equally, and that equal part is the smaller angle.

Answer: 55 degrees

4 · Reviewdoes it hold up?

If COD = 55 degrees, then BOC = 85 degrees. Check the line: 40 + 85 + 55 = 180 degrees, and 85 is exactly 30 more than 55. Everything fits.

Another way: Convert to algebra (tool 13): let COD = x, then BOC = x + 30, and 40 + (x + 30) + x = 180, giving 2x = 110, x = 55 degrees.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight-line total and the degree difference to find COD.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the rays OB, OC and the line AD through point O.

💡Takeaway. Take the known angle off the 180-degree line, set aside the extra, then share what is left in two - all Grade 4 angle adding and subtracting!

Variant 7 medium

Angle $\angle BOC$ is $30^\circ$ larger than angle $\angle COD$ . Using the figure, find the measure of angle $\angle COD$ .

[Figure] Straight line $AD$ passes through point $O$ , and two rays $OB$ and $OC$ extend upward from $O$ . Starting from the left end of the line, the angles $\angle AOB = 70^\circ$ , $\angle BOC$ , and $\angle COD$ lie in order, so the three angles together form the straight angle $180^\circ$ .

Your answer

Show solution

1 · Understandwhat's really being asked

On straight line AD through point O, rays OB and OC go upward. From the left, the angles AOB = 70 degrees, BOC, and COD line up and together make the straight angle of 180 degrees. Also angle BOC is 30 degrees bigger than angle COD. Find angle COD.

Givens

Angle AOB = 70 degrees.
Angles AOB, BOC, COD sit in order along straight line AD, so they add to 180 degrees.
Angle BOC = angle COD + 30 degrees.

Unknowns

The measure of angle COD.

Constraints

Angles on a straight line add to 180 degrees.
BOC is exactly 30 degrees larger than COD.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

First peel off the known 70 degrees to find how much BOC and COD share. Then, since BOC is 30 more than COD, take that extra off the shared total and split the rest into the two equal parts; guess-and-check confirms COD.

3 · Execute2 carry out the plan

1Find the total for BOC and COD

#7 Identify Subproblems 4.MD.C.7

The three angles fill the 180-degree straight line, so BOC plus COD is the part left after the known angle.

\angle BOC + \angle COD = 180^\circ - 70^\circ = 110^\circ

A flat line is 180 degrees; remove the known piece and the other two pieces share the rest.

2Remove the extra 30 degrees and split

#6 Guess and Check 4.MD.C.7

BOC is COD plus 30. If we set aside that extra 30 degrees from the 110, the leftover 80 degrees splits into two equal COD-sized parts. Half of 80 is the measure of COD.

(110^\circ - 30^\circ) \div 2 = 80^\circ \div 2 = 40^\circ

Two parts differ by a fixed amount; take the extra away so they match, share equally, and that equal part is the smaller angle.

Answer: 40 degrees

4 · Reviewdoes it hold up?

If COD = 40 degrees, then BOC = 70 degrees. Check the line: 70 + 70 + 40 = 180 degrees, and 70 is exactly 30 more than 40. Everything fits.

Another way: Convert to algebra (tool 13): let COD = x, then BOC = x + 30, and 70 + (x + 30) + x = 180, giving 2x = 80, x = 40 degrees.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight-line total and the degree difference to find COD.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the rays OB, OC and the line AD through point O.

💡Takeaway. Take the known angle off the 180-degree line, set aside the extra, then share what is left in two - all Grade 4 angle adding and subtracting!

Variant 8 medium

Angle $\angle BOC$ is $0^\circ$ larger than angle $\angle COD$ . Using the figure, find the measure of angle $\angle COD$ .

[Figure] Straight line $AD$ passes through point $O$ , and two rays $OB$ and $OC$ extend upward from $O$ . Starting from the left end of the line, the angles $\angle AOB = 40^\circ$ , $\angle BOC$ , and $\angle COD$ lie in order, so the three angles together form the straight angle $180^\circ$ .

Your answer

Show solution

1 · Understandwhat's really being asked

On straight line AD through point O, rays OB and OC go upward. From the left, the angles AOB = 40 degrees, BOC, and COD line up and together make the straight angle of 180 degrees. Also angle BOC is 0 degrees bigger than angle COD. Find angle COD.

Givens

Angle AOB = 40 degrees.
Angles AOB, BOC, COD sit in order along straight line AD, so they add to 180 degrees.
Angle BOC = angle COD + 0 degrees.

Unknowns

The measure of angle COD.

Constraints

Angles on a straight line add to 180 degrees.
BOC is exactly 0 degrees larger than COD.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

First peel off the known 40 degrees to find how much BOC and COD share. Then, since BOC is 0 more than COD, take that extra off the shared total and split the rest into the two equal parts; guess-and-check confirms COD.

3 · Execute2 carry out the plan

1Find the total for BOC and COD

#7 Identify Subproblems 4.MD.C.7

The three angles fill the 180-degree straight line, so BOC plus COD is the part left after the known angle.

\angle BOC + \angle COD = 180^\circ - 40^\circ = 140^\circ

A flat line is 180 degrees; remove the known piece and the other two pieces share the rest.

2Remove the extra 0 degrees and split

#6 Guess and Check 4.MD.C.7

BOC is COD plus 0. If we set aside that extra 0 degrees from the 140, the leftover 140 degrees splits into two equal COD-sized parts. Half of 140 is the measure of COD.

(140^\circ - 0^\circ) \div 2 = 140^\circ \div 2 = 70^\circ

Two parts differ by a fixed amount; take the extra away so they match, share equally, and that equal part is the smaller angle.

Answer: 70 degrees

4 · Reviewdoes it hold up?

If COD = 70 degrees, then BOC = 70 degrees. Check the line: 40 + 70 + 70 = 180 degrees, and 70 is exactly 0 more than 70. Everything fits.

Another way: Convert to algebra (tool 13): let COD = x, then BOC = x + 0, and 40 + (x + 0) + x = 180, giving 2x = 140, x = 70 degrees.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight-line total and the degree difference to find COD.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the rays OB, OC and the line AD through point O.

💡Takeaway. Take the known angle off the 180-degree line, set aside the extra, then share what is left in two - all Grade 4 angle adding and subtracting!

Variant 9 medium

Angle $\angle BOC$ is $40^\circ$ larger than angle $\angle COD$ . Using the figure, find the measure of angle $\angle COD$ .

[Figure] Straight line $AD$ passes through point $O$ , and two rays $OB$ and $OC$ extend upward from $O$ . Starting from the left end of the line, the angles $\angle AOB = 30^\circ$ , $\angle BOC$ , and $\angle COD$ lie in order, so the three angles together form the straight angle $180^\circ$ .

Your answer

Show solution

1 · Understandwhat's really being asked

On straight line AD through point O, rays OB and OC go upward. From the left, the angles AOB = 30 degrees, BOC, and COD line up and together make the straight angle of 180 degrees. Also angle BOC is 40 degrees bigger than angle COD. Find angle COD.

Givens

Angle AOB = 30 degrees.
Angles AOB, BOC, COD sit in order along straight line AD, so they add to 180 degrees.
Angle BOC = angle COD + 40 degrees.

Unknowns

The measure of angle COD.

Constraints

Angles on a straight line add to 180 degrees.
BOC is exactly 40 degrees larger than COD.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

First peel off the known 30 degrees to find how much BOC and COD share. Then, since BOC is 40 more than COD, take that extra off the shared total and split the rest into the two equal parts; guess-and-check confirms COD.

3 · Execute2 carry out the plan

1Find the total for BOC and COD

#7 Identify Subproblems 4.MD.C.7

The three angles fill the 180-degree straight line, so BOC plus COD is the part left after the known angle.

\angle BOC + \angle COD = 180^\circ - 30^\circ = 150^\circ

A flat line is 180 degrees; remove the known piece and the other two pieces share the rest.

2Remove the extra 40 degrees and split

#6 Guess and Check 4.MD.C.7

BOC is COD plus 40. If we set aside that extra 40 degrees from the 150, the leftover 110 degrees splits into two equal COD-sized parts. Half of 110 is the measure of COD.

(150^\circ - 40^\circ) \div 2 = 110^\circ \div 2 = 55^\circ

Two parts differ by a fixed amount; take the extra away so they match, share equally, and that equal part is the smaller angle.

Answer: 55 degrees

4 · Reviewdoes it hold up?

If COD = 55 degrees, then BOC = 95 degrees. Check the line: 30 + 95 + 55 = 180 degrees, and 95 is exactly 40 more than 55. Everything fits.

Another way: Convert to algebra (tool 13): let COD = x, then BOC = x + 40, and 30 + (x + 40) + x = 180, giving 2x = 110, x = 55 degrees.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight-line total and the degree difference to find COD.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the rays OB, OC and the line AD through point O.

💡Takeaway. Take the known angle off the 180-degree line, set aside the extra, then share what is left in two - all Grade 4 angle adding and subtracting!

Variant 10 medium

Angle $\angle BOC$ is $24^\circ$ larger than angle $\angle COD$ . Using the figure, find the measure of angle $\angle COD$ .

[Figure] Straight line $AD$ passes through point $O$ , and two rays $OB$ and $OC$ extend upward from $O$ . Starting from the left end of the line, the angles $\angle AOB = 36^\circ$ , $\angle BOC$ , and $\angle COD$ lie in order, so the three angles together form the straight angle $180^\circ$ .

Your answer

Show solution

1 · Understandwhat's really being asked

On straight line AD through point O, rays OB and OC go upward. From the left, the angles AOB = 36 degrees, BOC, and COD line up and together make the straight angle of 180 degrees. Also angle BOC is 24 degrees bigger than angle COD. Find angle COD.

Givens

Angle AOB = 36 degrees.
Angles AOB, BOC, COD sit in order along straight line AD, so they add to 180 degrees.
Angle BOC = angle COD + 24 degrees.

Unknowns

The measure of angle COD.

Constraints

Angles on a straight line add to 180 degrees.
BOC is exactly 24 degrees larger than COD.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #7 Identify Subproblems

First peel off the known 36 degrees to find how much BOC and COD share. Then, since BOC is 24 more than COD, take that extra off the shared total and split the rest into the two equal parts; guess-and-check confirms COD.

3 · Execute2 carry out the plan

1Find the total for BOC and COD

#7 Identify Subproblems 4.MD.C.7

The three angles fill the 180-degree straight line, so BOC plus COD is the part left after the known angle.

\angle BOC + \angle COD = 180^\circ - 36^\circ = 144^\circ

A flat line is 180 degrees; remove the known piece and the other two pieces share the rest.

2Remove the extra 24 degrees and split

#6 Guess and Check 4.MD.C.7

BOC is COD plus 24. If we set aside that extra 24 degrees from the 144, the leftover 120 degrees splits into two equal COD-sized parts. Half of 120 is the measure of COD.

(144^\circ - 24^\circ) \div 2 = 120^\circ \div 2 = 60^\circ

Two parts differ by a fixed amount; take the extra away so they match, share equally, and that equal part is the smaller angle.

Answer: 60 degrees

4 · Reviewdoes it hold up?

If COD = 60 degrees, then BOC = 84 degrees. Check the line: 36 + 84 + 60 = 180 degrees, and 84 is exactly 24 more than 60. Everything fits.

Another way: Convert to algebra (tool 13): let COD = x, then BOC = x + 24, and 36 + (x + 24) + x = 180, giving 2x = 120, x = 60 degrees.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight-line total and the degree difference to find COD.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the rays OB, OC and the line AD through point O.

💡Takeaway. Take the known angle off the 180-degree line, set aside the extra, then share what is left in two - all Grade 4 angle adding and subtracting!

Set up an equation with one unknown angle · 10 practice problems

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