Explore · MATH-5B-C11-L07-OPEN
A quadrilateral has angles 91°, 111°, and 61°. Find the fourth.
Singapore Math 5B · Original independent course
Angles of Quadrilaterals — Part 1
Watch for: Applying a triangle’s 180° angle sum to a figure with four sides.. Explain what the quantities mean before calculating.
Opening task
Prerequisite check
Before instruction · MATH-5B-C11-L07-PREREQ
What is 180° + 180°?
Show worked answer (after attempting)
Answer: 360°
Two triangle angle sums make 360 degrees.
Understand the idea
Derive and use the 360° interior-angle sum by dividing a simple quadrilateral into two triangles. The essential earlier idea is: Triangle interior angles and straight angles.. The model below will always use the actual quantities from the worked problem. Check applying a triangle’s 180° angle sum to a figure with four sides when you explain your answer.
Three worked examples
Worked example 1 · MATH-5B-C11-L07-Q01
A quadrilateral has angles 90°, 110°, and 60°. Find the fourth.
Solution: 100°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 260° to obtain 100°.
Worked example 2 · MATH-5B-C11-L07-Q02
A four-sided picture frame has three measured angles [72, 108, 88]. What angle is missing?
Solution: 92°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 268° to obtain 92°.
Worked example 3 · MATH-5B-C11-L07-Q03
Explain using two triangles why the fourth angle with other angles [95, 85, 105] is 75°.
Solution: 75°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 285° to obtain 75°.
Check understanding
1 · MATH-5B-C11-L07-Q04
A quadrilateral has angles 91°, 111°, and 61°. Find the fourth.
Show worked answer (after attempting)
Answer: 97°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 263° to obtain 97°.
2 · MATH-5B-C11-L07-Q05
A four-sided picture frame has three measured angles [73, 109, 89]. What angle is missing?
Show worked answer (after attempting)
Answer: 89°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 271° to obtain 89°.
3 · MATH-5B-C11-L07-Q06
Explain using two triangles why the fourth angle with other angles [96, 86, 106] is 72°.
Show worked answer (after attempting)
Answer: 72°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 288° to obtain 72°.
Guided practice
1 · MATH-5B-C11-L07-Q07
A quadrilateral has angles 92°, 112°, and 62°. Find the fourth.
Show worked answer (after attempting)
Answer: 94°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 266° to obtain 94°.
2 · MATH-5B-C11-L07-Q08
A four-sided picture frame has three measured angles [74, 110, 90]. What angle is missing?
Show worked answer (after attempting)
Answer: 86°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 274° to obtain 86°.
3 · MATH-5B-C11-L07-Q09
Explain using two triangles why the fourth angle with other angles [97, 87, 107] is 69°.
Show worked answer (after attempting)
Answer: 69°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 291° to obtain 69°.
4 · MATH-5B-C11-L07-Q10
A quadrilateral has angles 93°, 113°, and 63°. Find the fourth.
Show worked answer (after attempting)
Answer: 91°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 269° to obtain 91°.
Independent practice
1 · MATH-5B-C11-L07-Q11
A four-sided picture frame has three measured angles [75, 111, 91]. What angle is missing?
Show worked answer (after attempting)
Answer: 83°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 277° to obtain 83°.
2 · MATH-5B-C11-L07-Q12
Explain using two triangles why the fourth angle with other angles [98, 88, 108] is 66°.
Show worked answer (after attempting)
Answer: 66°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 294° to obtain 66°.
3 · MATH-5B-C11-L07-Q13
A quadrilateral has angles 94°, 114°, and 64°. Find the fourth.
Show worked answer (after attempting)
Answer: 88°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 272° to obtain 88°.
4 · MATH-5B-C11-L07-Q14
A four-sided picture frame has three measured angles [76, 112, 92]. What angle is missing?
Show worked answer (after attempting)
Answer: 80°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 280° to obtain 80°.
5 · MATH-5B-C11-L07-Q15
Explain using two triangles why the fourth angle with other angles [99, 89, 109] is 63°.
Show worked answer (after attempting)
Answer: 63°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 297° to obtain 63°.
6 · MATH-5B-C11-L07-Q16
A quadrilateral has angles 95°, 115°, and 65°. Find the fourth.
Show worked answer (after attempting)
Answer: 85°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 275° to obtain 85°.
7 · MATH-5B-C11-L07-Q17
A four-sided picture frame has three measured angles [77, 113, 93]. What angle is missing?
Show worked answer (after attempting)
Answer: 77°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 283° to obtain 77°.
8 · MATH-5B-C11-L07-Q18
Explain using two triangles why the fourth angle with other angles [100, 90, 110] is 60°.
Show worked answer (after attempting)
Answer: 60°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 300° to obtain 60°.
Challenge and error analysis
1 · MATH-5B-C11-L07-ERR1
The assigned question is: “A four-sided picture frame has three measured angles [80, 116, 96]. What angle is missing?” A learner wrote: “I used a total of 180 degrees for the four interior angles rather than 360 degrees, producing 68.1°.” Identify a concrete invalid step or assumption. Work out the correct result and explain why it is valid.
Show worked answer (after attempting)
Answer: Correct result: 68°
The attempted reasoning or reported result does not satisfy the specified operation, units or geometric conditions. Correct mathematical result: 68°. Recompute from the given data: A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 292° to obtain 68°.
2 · MATH-5B-C11-L07-ERR2
The assigned question is: “Explain using two triangles why the fourth angle with other angles [103, 93, 113] is 51°.” A learner wrote: “I used a total of 180 degrees for the four interior angles rather than 360 degrees, producing 52°.” Identify a concrete invalid step or assumption. Work out the correct result and explain why it is valid.
Show worked answer (after attempting)
Answer: Correct result: 51°
The attempted reasoning or reported result does not satisfy the specified operation, units or geometric conditions. Correct mathematical result: 51°. Recompute from the given data: A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 309° to obtain 51°.
Exit check
Exit · MATH-5B-C11-L07-Q19
A quadrilateral has angles 96°, 116°, and 66°. Find the fourth.
Show worked answer (after attempting)
Answer: 82°
A quadrilateral splits into two triangles: 180° + 180° = 360°. Subtract 278° to obtain 82°.