Year 7 Mathematics · AC9M7M05

Triangle and Polygon Angle Sums

demonstrate that the interior angle sum of a triangle in the plane is 180° and apply this to determine the interior angle sum of other shapes and the size of unknown…

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Learning goalsSay it simply

We are learning to demonstrate the triangle angle sum and use triangulation to calculate polygon interior sums and unknown angles.

In Euclidean geometry, the three interior angles of every triangle sum to 180°. Rearranging the three corner angles into a straight angle gives an empirical demonstration; a parallel line through a vertex supports a deductive demonstration using alternate angles.

A diagonal from one vertex can triangulate a convex polygon without gaps or overlaps. A quadrilateral forms 2 triangles, a pentagon 3 and an n-sided polygon n−2, so its interior angle sum is (n−2)×180°.

Use the appropriate triangle or polygon interior sum with stated equal-angle properties to find unknown angles. Check that every triangle contributes exactly 180° and that the reconstructed polygon angles total the calculated sum.

Success criteria

  • I can demonstrate why a Euclidean triangle’s interior angles sum to 180°.
  • I can triangulate a convex polygon and calculate its interior angle sum.
  • I can determine unknown triangle and polygon angles and justify an independent check.
Key conceptTeach from the board

Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.

Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.

Example 1

Demonstrate the triangle angle sum

  1. Draw triangle ABC and extend a straight line through vertex A parallel to side BC.
  2. Use alternate angles to copy angles B and C beside angle A on the straight line.
  3. Observe that the three adjacent angles fill the straight angle without gaps or overlaps.
  4. Conclude that A+B+C=180° for every Euclidean triangle.

Final answer: The interior angles of a Euclidean triangle sum to 180°.

Check: For example, 50°+60°+70°=180°.

Example 2

Triangulate a pentagon to find an unknown angle

  1. Draw diagonals from one vertex of the convex pentagon to form 3 non-overlapping triangles.
  2. Calculate the pentagon interior sum as 3×180°=540°.
  3. Add the four known angles: 95°+110°+120°+105°=430°.
  4. Subtract to find x=540°−430°=110°.

Final answer: The unknown pentagon angle is 110°.

Check: The five angles total 95°+110°+120°+105°+110°=540°.

Example 3

Application problem 1

Problem: A pentagon has four interior angles 112°, 95°, 130° and 108°. Find the fifth angle by first determining the pentagon's interior-angle sum.

  1. Plan: Represent the information first, then calculate, interpret and independently check the result.
  2. Work: A pentagon has (5−2)×180° = 540°. The known angles total 445°, so the fifth angle is 95°.
  3. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Final answer: A pentagon has (5−2)×180° = 540°. The known angles total 445°, so the fifth angle is 95°.

Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Application problem 2

Problem: A convex hexagon has five interior angles of 110°, 125°, 130°, 140° and 95°. Find the sixth angle and justify the total using triangulation.

  1. Plan: Use 4×180° for the hexagon sum, then subtract the five known angles.
  2. Work: The sixth angle is 120° because a hexagon forms 4 triangles with total 720°, while the five known angles total 600°.
  3. Interpret: Triangulating the hexagon gives (6−2)×180°=720°; the known angles total 600°, so the sixth angle is 720°−600°=120°.

Final answer: The sixth angle is 120° because a hexagon forms 4 triangles with total 720°, while the five known angles total 600°.

Check: Triangulating the hexagon gives (6−2)×180°=720°; the known angles total 600°, so the sixth angle is 720°−600°=120°.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7M05 - Triangle and Polygon Angle Sums
Example 1

Example 1 Demonstrate the triangle angle sum Draw triangle ABC and extend a straight line through vertex A parallel to side BC. Use alternate angles to copy angles B and C beside angle A on the straight line. Observe that the three adjacent angles fill the straight angle without gaps or overlaps. Conclude that A+B+C=180° for every Euclidean triangle. Final answer: The interior angles of a Euclidean triangle sum to 180°. Check: For example, 50°+60°+70°=180°.

Example 2

Example 2 Triangulate a pentagon to find an unknown angle Draw diagonals from one vertex of the convex pentagon to form 3 non-overlapping triangles. Calculate the pentagon interior sum as 3×180°=540°. Add the four known angles: 95°+110°+120°+105°=430°. Subtract to find x=540°−430°=110°. Final answer: The unknown pentagon angle is 110°. Check: The five angles total 95°+110°+120°+105°+110°=540°.

Example 3

Example 3 Application problem 1 Problem: A pentagon has four interior angles 112°, 95°, 130° and 108°. Find the fifth angle by first determining the pentagon's interior-angle sum. Plan: Represent the information first, then calculate, interpret and independently check the result. Work: A pentagon has (5−2)×180° = 540°. The known angles total 445°, so the fifth angle is 95°. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills. Final answer: A pentagon has (5−2)×180° = 540°. The known angles total 445°, so the fifth angle is 95°. Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Example 4 Application problem 2 Problem: A convex hexagon has five interior angles of 110°, 125°, 130°, 140° and 95°. Find the sixth angle and justify the total using triangulation. Plan: Use 4×180° for the hexagon sum, then subtract the five known angles. Work: The sixth angle is 120° because a hexagon forms 4 triangles with total 720°, while the five known angles total 600°. Interpret: Triangulating the hexagon gives (6−2)×180°=720°; the known angles total 600°, so the sixth angle is 720°−600°=120°. Final answer: The sixth angle is 120° because a hexagon forms 4 triangles with total 720°, while the five known angles total 600°. Check: Triangulating the hexagon gives (6−2)×180°=720°; the known angles total 600°, so the sixth angle is 720°−600°=120°.

Curriculum examplesCopied content

Content description: demonstrate that the interior angle sum of a triangle in the plane is 180° and apply this to determine the interior angle sum of other shapes and the size of unknown angles.

Questions and answersWith answers

Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.

  1. 1. Find the third angle of a triangle with angles 50° and 60°.

    Check answer

    Answer: 70°.

    Hint: Subtract both known angles from 180°.

    Why: A triangle totals 180°, so 180°−50°−60°=70°.

  2. 2. A quadrilateral is split by one diagonal into two triangles. Find its interior angle sum.

    Check answer

    Answer: 360°.

    Hint: Multiply the 2 triangles by 180°.

    Why: The two non-overlapping triangles contribute 2×180°=360° in total.

  3. 3. How many triangles are formed when diagonals are drawn from one vertex of a convex pentagon?

    Check answer

    Answer: 3 triangles.

    Hint: Use n−2 with n=5.

    Why: A pentagon triangulated from one vertex forms 5−2=3 triangles.

  4. 4. A quadrilateral has interior angles 88°, 104°, 97° and x. Find x.

    Check answer

    Answer: x=71°.

    Hint: Subtract the three known angles from 360°.

    Why: The quadrilateral sum is 360°, and 360°−(88°+104°+97°)=71°.

  5. 5. Use triangulation to find the interior angle sum of a convex hexagon.

    Check answer

    Answer: 720°.

    Hint: Multiply the hexagon’s 6−2 triangles by 180°.

    Why: A convex hexagon forms 4 triangles, so its interior angle sum is 4×180°=720°.

  6. 6. A polygon triangulated from one vertex forms 7 triangles. How many sides does it have, and what is its interior angle sum?

    Check answer

    Answer: It has 9 sides and an interior angle sum of 1260°.

    Hint: Solve n−2=7, then multiply 7 by 180°.

    Why: Seven triangles mean n=9 sides, and their angle sums combine to 7×180°=1260°.

  7. 7. A student says the interior angle sum of an n-sided polygon is n×180°. Explain and correct the error.

    Check answer

    Answer: The rule is (n−2)×180° because triangulation from one vertex forms n−2 triangles.

    Hint: Draw diagonals from one vertex and count the triangles, not the vertices.

    Why: Triangulation shows there are n−2 triangles, so the correct sum is (n−2)×180°, not n×180°.

  8. 8. A quadrilateral has interior angles x, x+20°, 2x and 80°. Find x and all four angles, then check the polygon sum.

    Check answer

    Answer: x=65°; the angles are 65°, 85°, 130° and 80°, which total 360°.

    Hint: Set the four expressions equal to the quadrilateral sum of 360°.

    Why: The equation x+(x+20)+2x+80=360 gives 4x+100=360 and x=65°, so the four angles are 65°, 85°, 130° and 80°.

  9. 9. A convex hexagon has five interior angles of 110°, 125°, 130°, 140° and 95°. Find the sixth angle and justify the total using triangulation.

    Check answer

    Answer: The sixth angle is 120° because a hexagon forms 4 triangles with total 720°, while the five known angles total 600°.

    Hint: Use 4×180° for the hexagon sum, then subtract the five known angles.

    Why: Triangulating the hexagon gives (6−2)×180°=720°; the known angles total 600°, so the sixth angle is 720°−600°=120°.

  10. 10. A pentagon has four interior angles 112°, 95°, 130° and 108°. Find the fifth angle by first determining the pentagon's interior-angle sum.

    Check answer

    Answer: A pentagon has (5−2)×180° = 540°. The known angles total 445°, so the fifth angle is 95°.

    Hint: Represent the information first, then calculate, interpret and independently check the result.

    Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Practice and reviewReady for practice

Common mistake: Every polygon has an interior angle sum of 180°.

Correction: Only a triangle has a 180° sum; triangulate the polygon and multiply 180° by the number of triangles.

Common mistake: An n-sided polygon can be split into n triangles from one vertex.

Correction: A convex n-sided polygon forms n−2 non-overlapping triangles from one vertex.

Common mistake: A diagonal creates extra angle measure.

Correction: A diagonal only partitions existing interior angles, so the triangle sums combine to the original polygon sum.

Curriculum alignmentStart here

We are learning to demonstrate the triangle angle sum and use triangulation to calculate polygon interior sums and unknown angles.

In Euclidean geometry, the three interior angles of every triangle sum to 180°. Rearranging the three corner angles into a straight angle gives an empirical demonstration; a parallel line through a vertex supports a deductive demonstration using alternate angles.

A diagonal from one vertex can triangulate a convex polygon without gaps or overlaps. A quadrilateral forms 2 triangles, a pentagon 3 and an n-sided polygon n−2, so its interior angle sum is (n−2)×180°.

Use the appropriate triangle or polygon interior sum with stated equal-angle properties to find unknown angles. Check that every triangle contributes exactly 180° and that the reconstructed polygon angles total the calculated sum.

Success criteria

  • I can demonstrate why a Euclidean triangle’s interior angles sum to 180°.
  • I can triangulate a convex polygon and calculate its interior angle sum.
  • I can determine unknown triangle and polygon angles and justify an independent check.
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