Triangle and polygon angle sums practice questions | Year 7 Mathematics

Vertex	Interior angle
\(A_{1}\)	\(x\)
\(A_{2}\)	\(105^\circ\)
\(A_{3}\)	\(112^\circ\)
\(A_{4}\)	\(98^\circ\)
\(A_{5}\)	\(125^\circ\)

Vertex	Interior angle
\(A_{1}\)	\(x\)
\(A_{2}\)	\(112^\circ\)
\(A_{3}\)	\(124^\circ\)
\(A_{4}\)	\(108^\circ\)
\(A_{5}\)	\(119^\circ\)
\(A_{6}\)	\(127^\circ\)

Vertex	Interior angle
\(A_{1}\)	\(x\)
\(A_{2}\)	\(130^\circ\)
\(A_{3}\)	\(140^\circ\)
\(A_{4}\)	\(135^\circ\)
\(A_{5}\)	\(128^\circ\)
\(A_{6}\)	\(137^\circ\)
\(A_{7}\)	\(142^\circ\)
\(A_{8}\)	\(138^\circ\)

Vertex	Interior angle
\(A_{1}\)	\(x\)
\(A_{2}\)	\(90^\circ\)
\(A_{3}\)	\(45^\circ\)
\(A_{4}\)	\(270^\circ\)
\(A_{5}\)	\(45^\circ\)

Vertex	Interior angle
\(A_{1}\)	\(x\)
\(A_{2}\)	\(80^\circ\)
\(A_{3}\)	\(105^\circ\)
\(A_{4}\)	\(240^\circ\)
\(A_{5}\)	\(70^\circ\)
\(A_{6}\)	\(95^\circ\)

Vertex	Interior angle
\(A_{1}\)	\(x\)
\(A_{2}\)	\(100^\circ\)
\(A_{3}\)	\(130^\circ\)
\(A_{4}\)	\(250^\circ\)
\(A_{5}\)	\(90^\circ\)
\(A_{6}\)	\(110^\circ\)
\(A_{7}\)	\(120^\circ\)

Common mistakes

Using one total for every shape

✕\(\displaystyle Every polygon has interior-angle sum 360°\)✓\(\displaystyle Use (n − 2) × 180° for a simple n-sided polygon\)

Different side counts give different numbers of interior triangles

Forgetting the second base angle

✕\(\displaystyle Subtract one base angle to find an isosceles apex\)✓\(\displaystyle Subtract both equal base angles from 180°\)

The triangle total includes three angles, not just the given one and the unknown

Choosing the adjacent angle in the exterior rule

✕\(\displaystyle An exterior angle equals an adjacent angle plus one remote angle\)✓\(\displaystyle It equals the sum of the two remote interior angles\)

The adjacent interior angle instead forms a straight angle with the exterior angle

Replacing a reflex interior angle

✕\(\displaystyle Use 120° instead of a stated 240° interior angle\)✓\(\displaystyle Include the full 240° interior measure\)

The smaller outside gap does not belong to the polygon interior sum

Difficulty levels

introUse the triangle total with acute, right and obtuse cases

standardUse equal base angles, quadrilateral totals and convex polygon unknowns

challengeUse exterior-angle relationships and include reflex angles in concave polygon calculations

Questions about this topic

Does the polygon need to be regular?+

No. The interior-angle sum depends on side count for a simple polygon, not equal sides or equal angles

Why is a quadrilateral total twice the triangle total?+

An inside diagonal partitions it into two triangles, and their angles together make the original four interior angles

Can I find one polygon angle by dividing the total by the number of sides?+

Only if all interior angles are equal. Otherwise subtract the sum of the known angles from the total

Does moving the triangle prove the angle-sum rule?+

The movement illustrates examples. The parallel-line construction, alternate interior equality and straight-angle total justify the rule

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 7 Mathematics, chapter 4: Geometry and measurement

In this chapter

Classifying shapes by their properties Angle facts and parallel lines Triangle and polygon angle sums Areas of rectangles, triangles and parallelograms Representing solids with nets and views Volumes of right prisms Radius diameter circumference and pi Model lengths, areas and volumes with ratios Translations, reflections and rotations with coordinates Algorithms for classifying shapes

Search

Triangle and polygon angle sums

Add the known interior angles before subtracting

Find an acute missing angle

Choose the triangle rule and construct the answer

Use the stated right angle in the sum

Calculate the second acute angle of a right triangle

A small known pair can leave an obtuse unknown

Include a known obtuse angle without changing the total

Locate the equal sides before equating angles

Share the remainder between two equal base angles

Remember the division after subtracting the apex

Subtract both equal base angles

Count the given base measure twice

Derive the constant angle of an equilateral triangle

Derive the exterior-angle rule instead of mixing the angles

Add the two remote interior measures

Select the exterior relationship and calculate

Work backwards from an exterior measure

Find the other remote interior angle

Build a quadrilateral total from two triangle totals

Add three unequal given angles before subtracting

Use the four-sided total, not the triangle total

Count triangles, not sides, in the total

Substitute the side count into the total formula

Count four triangles inside a hexagon

Keep the side count and triangle count distinct

Separate the polygon total from the sum of the givens

Calculate a missing angle in an irregular convex pentagon

Use all five known angles of the hexagon

Organise a longer known-angle sum

Keep the full interior angle at an inward corner

Include the 270-degree interior angle in the pentagon total

Use the supplied reflex angle without replacing it

Combine a seven-sided total with a reflex interior measure

Justify the total, add the givens, calculate and check

Common mistakes

Difficulty levels

Questions about this topic

In this chapter