Lesson Guide
Explanation
A line of symmetry folds a shape exactly onto itself, so every corner off the line lands on its partner, and those two angles are equal. A corner sitting on the fold has no partner and stands alone. A triangle with one line of symmetry therefore has one pair of equal angles and one odd one; a kite has one pair and two odd ones. The little tick marks on the sides show which sides are equal, and equal sides sit opposite equal angles.
Your Turn
How many pairs of equal angles does a shape with 5 corners and one line of symmetry have, if one corner is on the line? Check: 2 pairs.
Watch Out
The angle ON the fold is not part of a pair. Do not assume all the angles are equal.
Explanation
The angles of any triangle add to a hundred and eighty degrees — fold the three corners in and they make a straight line. In an isosceles triangle two of them are equal, so subtract the one you know and halve what is left: an apex of forty gives base angles of seventy. It works backwards too: base angles of sixty-five leave an apex of fifty. An equilateral triangle is the special case where all three are equal, sixty degrees each.
Your Turn
An isosceles triangle has an apex angle of 100°. Find each base angle. Check: (180 − 100) ÷ 2 = 40°.
Watch Out
Halve only the two equal angles. Halving 180 first, then subtracting, gives the wrong answer.
Explanation
The angles of any quadrilateral add to three hundred and sixty degrees. In a kite, the two angles on the fold line are the odd ones out and the other two are equal. Given a hundred and twenty at the top and sixty at the bottom, the equal pair is three hundred and sixty minus a hundred and eighty, halved: ninety each. Check by adding all four back to three hundred and sixty, which is the only check worth doing.
Your Turn
A kite has 100° at the top and 60° at the bottom. Find each of the other two angles. Check: (360 − 160) ÷ 2 = 100°.
Watch Out
In a kite the equal pair is the two angles OFF the fold line, not the two on it.
Explanation
Two questions about the same triangle need different work. If the pair is given — fifty and fifty — the odd angle is one subtraction, eighty. If the odd angle is given — eighty — the pair needs a subtraction and then a halving, fifty each. So look at the marks first: equal ticks on the sides, or the same letter on two angles, tells you which two match, and that decides whether you divide.
Your Turn
An isosceles triangle has base angles of 50°. What is the apex? Check: 180 − 100 = 80°.
Watch Out
Divide by 2 only when you are finding the PAIR. If the pair is already known, one subtraction is enough.
Explanation
The routine is short. Decide the total: a hundred and eighty for a triangle, three hundred and sixty for a quadrilateral. Decide whether the pair is what you know or what you want: if you want it, subtract then halve; if you know it, subtract only. A roof truss with a thirty-degree apex has base angles of seventy-five. Then add your answers back to the total, every time. Quiz and exercises now.
Your Turn
A roof truss is an isosceles triangle with an apex of 30°. Find each base angle. Check: (180 − 30) ÷ 2 = 75°.
Watch Out
Count the corners to pick the total. Four corners means 360°, even when the shape looks like two triangles.
📌 Key Points — this lesson in a nutshell
- A line of symmetry folds a shape onto itself, so it matches angles in pairs.
- The angles of any triangle add to 180°; of any quadrilateral, to 360°.
- An isosceles triangle has two equal sides and two equal base angles. Apex 40° → base angles (180 − 40) ÷ 2 = 70°.
- A kite has one pair of equal angles — the two the fold swaps. Find the pair first, then subtract from 360°.
Step 1: Introduction
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