PAPER · REASONING · PRACTICE

Triangle and polygon angles

Choose the correct angle sum, distinguish adjacent and remote angles, and count polygon sides.

40 minutes · Year 8, upper volume · text pp. 5–12 · foundation lessons 3–6

Textbook reference: Year 8, upper · Chapter 11: triangles · lessons 3–6

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the upper-volume plan

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Teacher reference Arrange in My Classroom

Lesson routePhase

4′Check prerequisites
5′Observe and predict
8′Worked example and guided attempt
15′Independent practice
5′Discuss and correct
3′Exit questions

Observe · explain

A question before practising

Triangulate a convex pentagon from one vertex, then turn around its boundary. Why do the two sums use different rules?

Organise the key idea

Triangle interior angles total 180°; a triangle exterior angle equals the two remote interior angles. A convex n-gon has interior sum (n−2)180° and exterior sum 360°.

Open the original chapter Open the interactive inquiry

Use the lesson question here for paper discussion; the original activity opens separately.

Worked example1

Two interior angles are 42° and 68°. Find the third angle with a reason.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

The three interior angles total 180°.
The missing angle is 180−42−68=70°.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Guided attempt2

A convex pentagon is divided from one vertex into triangles. Complete the count and find its interior-angle sum.

Number of triangles=5−___=___.
Interior sum=___×180°=___.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

A pentagon gives 5−2=3 triangles without overlap.
Its interior sum is 3×180°=540°.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Prerequisite3

An interior angle is 65°. Find its adjacent exterior angle.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

Adjacent interior and exterior angles form a straight angle.
180−65=115°.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Prerequisite4

A right triangle has one acute angle 32°. Find the other acute angle.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

The two acute angles total 90°.
90−32=58°.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Independent practice5

Two interior angles of a triangle are 37° and 76°. Find the remaining angle.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

Use the triangle interior-angle sum.
180−37−76=67°.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Independent practice6

The two remote interior angles to a triangle exterior angle are 48° and 57°. Find the exterior angle.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

An exterior angle equals the sum of the remote interior angles.
48+57=105°; the adjacent interior angle is 75°.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Independent practice7

Find the sum of the interior angles of a convex hexagon.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

A hexagon has 6−2=4 triangles in the vertex triangulation.
4×180°=720°.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Independent practice8

Every exterior angle of a regular convex polygon is 24°. How many sides does it have?

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

One exterior angle at each vertex, taken in the same turning direction, totals 360°.
The number of sides is 360/24=15.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Exit question9

A triangle interior angle is 112°. Find its adjacent exterior angle.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

Use the straight angle at that vertex.
180−112=68°.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Exit question10

What is the sum of one exterior angle at each vertex of a convex nonagon, all taken in the same direction?

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

Traversing a convex polygon makes one complete turn.
The exterior-angle sum is 360°, independent of the number of sides.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Further practice11

A triangle has angles x°,2x° and 3x°. Find x.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

x+2x+3x=180, so 6x=180.
x=30; the angles 30°,60°,90° are positive and total 180°.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Further practice12

An exterior angle is 121° and one remote interior angle is 54°. Find the other remote interior angle.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

The two remote interior angles total 121°.
The required angle is 121−54=67°; the adjacent interior angle is 59°.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Further practice13

Find the interior-angle sum of a convex octagon.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

Use (n−2)180° with n=8.
6×180°=1080°.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Further practice14

Find each interior angle of a regular convex decagon.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

The interior sum is (10−2)180°=1440°.
Divide by 10: each angle is 144°. Its exterior angle is 36°.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Extension15

A convex polygon has interior-angle sum 1980°. Find its number of sides.

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

(n−2)180=1980, so n−2=11.
Thus n=13, an integer at least 3.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Extension16

Can 20°,70°,100° be the interior angles of a triangle?

Yes: all are positive.
Yes: one angle is obtuse.
Only for an isosceles triangle.
No: their sum is 190°.

Triangle angles must be positive and total exactly 180°.
20+70+100=190, so the proposed triple is impossible.
Compare with: No: their sum is 190°.

Triangle angles must be positive and total exactly 180°.
20+70+100=190, so the proposed triple is impossible.

Check your work

Choose the statement justified by the stated conditions and explain why.

Preparation and assessment notes

Prerequisites

Straight angles, simple equations and the triangle angle-sum theorem.

Look for in students’ work

Use the worked model and guided attempt selectively, then the eight core questions. Keep further practice and extensions optional. Ask for a reason and an original-equation or expansion check.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.

Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.

Print edition

Textbook reference and authorship

Chapter-core practice after the mapped textbook lessons; these tasks do not replace every first-teaching period.

大湾区初中数学三维课堂（八年级上册）正文.pdf

Year 8, upper volume · text pp. 5–12 · foundation lessons 3–6

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Triangle and polygon angles

Year 8, upper volume · text pp. 5–12 · foundation lessons 3–6 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Two interior angles are 42° and 68°. Find the third angle with a reason.

Guided attempt2

A convex pentagon is divided from one vertex into triangles. Complete the count and find its interior-angle sum.

Number of triangles=5−___=___.
Interior sum=___×180°=___.

Prerequisite3

An interior angle is 65°. Find its adjacent exterior angle.

Prerequisite4

A right triangle has one acute angle 32°. Find the other acute angle.

Triangle and polygon angles

Year 8, upper volume · text pp. 5–12 · foundation lessons 3–6 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Two interior angles of a triangle are 37° and 76°. Find the remaining angle.

Independent practice6

The two remote interior angles to a triangle exterior angle are 48° and 57°. Find the exterior angle.

Independent practice7

Find the sum of the interior angles of a convex hexagon.

Independent practice8

Every exterior angle of a regular convex polygon is 24°. How many sides does it have?

Exit question9

A triangle interior angle is 112°. Find its adjacent exterior angle.

Exit question10

What is the sum of one exterior angle at each vertex of a convex nonagon, all taken in the same direction?

Triangle and polygon angles — Further practice

Year 8, upper volume · text pp. 5–12 · foundation lessons 3–6

Further practice11

A triangle has angles x°,2x° and 3x°. Find x.

Further practice12

An exterior angle is 121° and one remote interior angle is 54°. Find the other remote interior angle.

Further practice13

Find the interior-angle sum of a convex octagon.

Further practice14

Find each interior angle of a regular convex decagon.

Extension15

A convex polygon has interior-angle sum 1980°. Find its number of sides.

Extension16

Can 20°,70°,100° be the interior angles of a triangle?

Yes: all are positive.
Yes: one angle is obtuse.
Only for an isosceles triangle.
No: their sum is 190°.

Triangle and polygon angles — Teacher answers

Year 8, upper volume · text pp. 5–12 · foundation lessons 3–6

Use the worked model and guided attempt selectively, then the eight core questions. Keep further practice and extensions optional. Ask for a reason and an original-equation or expansion check.

Worked example1

Two interior angles are 42° and 68°. Find the third angle with a reason.

Reference answer

70 °.

180-42-68

The three interior angles total 180°.
The missing angle is 180−42−68=70°.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Guided attempt2

A convex pentagon is divided from one vertex into triangles. Complete the count and find its interior-angle sum.

Number of triangles=5−___=___.
Interior sum=___×180°=___.

Reference answer

540 °.

(5-2)*180

A pentagon gives 5−2=3 triangles without overlap.
Its interior sum is 3×180°=540°.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Prerequisite3

An interior angle is 65°. Find its adjacent exterior angle.

Reference answer

115 °.

180-65

Adjacent interior and exterior angles form a straight angle.
180−65=115°.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Prerequisite4

A right triangle has one acute angle 32°. Find the other acute angle.

Reference answer

58 °.

180-90-32

The two acute angles total 90°.
90−32=58°.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Independent practice5

Two interior angles of a triangle are 37° and 76°. Find the remaining angle.

Reference answer

67 °.

180-37-76

Use the triangle interior-angle sum.
180−37−76=67°.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Independent practice6

The two remote interior angles to a triangle exterior angle are 48° and 57°. Find the exterior angle.

Reference answer

105 °.

48+57

An exterior angle equals the sum of the remote interior angles.
48+57=105°; the adjacent interior angle is 75°.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Independent practice7

Find the sum of the interior angles of a convex hexagon.

Reference answer

720 °.

(6-2)*180

A hexagon has 6−2=4 triangles in the vertex triangulation.
4×180°=720°.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Independent practice8

Every exterior angle of a regular convex polygon is 24°. How many sides does it have?

Reference answer

15.

360/24

One exterior angle at each vertex, taken in the same turning direction, totals 360°.
The number of sides is 360/24=15.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Exit question9

A triangle interior angle is 112°. Find its adjacent exterior angle.

Reference answer

68 °.

180-112

Use the straight angle at that vertex.
180−112=68°.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Exit question10

What is the sum of one exterior angle at each vertex of a convex nonagon, all taken in the same direction?

Reference answer

360 °.

360

Traversing a convex polygon makes one complete turn.
The exterior-angle sum is 360°, independent of the number of sides.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Further practice11

A triangle has angles x°,2x° and 3x°. Find x.

Reference answer

30.

180/6

x+2x+3x=180, so 6x=180.
x=30; the angles 30°,60°,90° are positive and total 180°.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Further practice12

An exterior angle is 121° and one remote interior angle is 54°. Find the other remote interior angle.

Reference answer

67 °.

121-54

The two remote interior angles total 121°.
The required angle is 121−54=67°; the adjacent interior angle is 59°.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Further practice13

Find the interior-angle sum of a convex octagon.

Reference answer

1080 °.

(8-2)*180

Use (n−2)180° with n=8.
6×180°=1080°.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Further practice14

Find each interior angle of a regular convex decagon.

Reference answer

144 °.

((10-2)*180)/10

The interior sum is (10−2)180°=1440°.
Divide by 10: each angle is 144°. Its exterior angle is 36°.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Extension15

A convex polygon has interior-angle sum 1980°. Find its number of sides.

Reference answer

13.

1980/180+2

(n−2)180=1980, so n−2=11.
Thus n=13, an integer at least 3.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Extension16

Can 20°,70°,100° be the interior angles of a triangle?

Yes: all are positive.
Yes: one angle is obtuse.
Only for an isosceles triangle.
No: their sum is 190°.

Reference answer

No: their sum is 190°.

Triangle angles must be positive and total exactly 180°.
20+70+100=190, so the proposed triple is impossible.

Choose the statement justified by the stated conditions and explain why.