Measure and draw angles
Name an angle by its vertex, align a protractor and choose the scale starting at zero.
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The middle letter names the vertex
In ∠ ABC, B is the vertex. The arms follow BA and BC. You can name the same opening ∠ CBA. Writing ∠ BAC instead moves the vertex to A.
A small label such as ∠ a names the opening marked by a in a diagram. Follow that mark rather than assuming it means an entire turn.
Choose the right opening at a shared vertex
Two openings at B
Each small letter marks one opening. Read its two arms before writing a name.
View figure data
| Point | x | y |
|---|---|---|
| Vertex B | 0 | 0 |
| Arm endpoint A | 8 | 0 |
| Arm endpoint C | 4 | 6 |
| Arm endpoint D | -5 | 4 |
The arc marked a runs from BA to BC. Start at A, pass through the vertex B and finish at C: ∠ ABC. Starting at C gives ∠ CBA, the same opening. Both names keep B in the middle.
The small-letter name is ∠ a. On a keyboard you can type angle a or angle ABC when an angle symbol is difficult to enter. Use capitals for the named points and the small letter shown inside the arc for the opening.
The neighbouring opening b has arms BC and BD, so it is ∠ CBD or ∠ DBC. The name ∠ ABD spans both marked openings, not just a. Writing only ∠ B would not tell someone which opening you meant here.
Check your understanding 1 Name an opening after the drawing is turned
Read the labels at Q
Identify the shared vertex and follow the arms of each marked opening.
View figure data
| Point | x | y |
|---|---|---|
| Vertex Q | 1 | -1 |
| Arm endpoint P | -5 | 1 |
| Arm endpoint R | 0 | -6 |
| Arm endpoint S | 6 | -3 |
Name the opening y using three capital point letters. Then name ∠ PQR using the small letter marking its opening. Explain why Q stays in the middle of a three-point name.
Show answer
The opening y is ∠ RQS or ∠ SQR: its arms are QR and QS. The opening ∠ PQR is ∠ x: its arms are QP and QR. Q is the meeting point of the two arms, so it is the middle letter. Turning the drawing does not change these names.
A right angle is 90 degrees
Angles are measured in degrees, written °. A right angle is 90°. Before measuring, estimate whether the angle is smaller or greater than 90°. This helps you detect a wrong protractor scale.
To measure ∠ ABC:
- Place the protractor’s centre exactly on B.
- Align its baseline with arm BA.
- Find the zero on the side of BA and follow that scale.
- Read where arm BC crosses it, extending the arm with a ruler if needed.
If an opening is clearly smaller than a right angle and the two scales show 40° and 140°, choose 40°. The correct scale begins at zero on the starting arm.
Position the tool before reading
In the drawing below, A and C are on the two arms and B is their meeting point. Identify ∠ABC, then find the centre and the zero you need on the protractor.
Place two points on the canvas: the protractor’s centre first, then a point in the direction of outer zero. Finish positioning to see the tool. Move its dots to adjust it. You can also type the two positions or open the drawing board for more space.
The small ticks mark 1°. Longer ticks mark 5° and 10°, and the printed numbers are 30° apart. Choose the scale that starts at zero on an arm; do not jump to a number just because it is nearby.
Position the protractor and measure the smaller opening ∠ABC. This activity has no score. You can use paper and a real protractor, or the canvas tool.
Coordinates increase to the right and down the page. Lengths use equal coordinate scales.
| Object | Values | Given or drawn |
|---|---|---|
| Given arm BA | (100, 100) → (180, 100) | Given |
| Given arm BC | (100, 100) → (140, 31) | Given |
| B | (100, 100), included | Given |
| A | (180, 100), included | Given |
| C | (140, 31), included | Given |
The check looks for the centre at B, a baseline along an arm, both arms under the semicircle, and a whole-degree reading within 1°. This does not certify physical instrument use. You can also draw on paper and compare with the public answer.
Show solution
Coordinates increase to the right and down the page. Lengths use equal coordinate scales.
| Object | Values | Given or drawn |
|---|---|---|
| Given arm BA | (100, 100) → (180, 100) | Given |
| Given arm BC | (100, 100) → (140, 31) | Given |
| B | (100, 100), included | Given |
| A | (180, 100), included | Given |
| C | (140, 31), included | Given |
| Protractor | Protractor centre (100, 100); direction towards outer 0 (160, 100); radius 60. Small ticks are 1°. Labels are 30° apart. The outer scale increases from 0 to 180 and the inner scale decreases from 180 to 0. | Drawn |
About 60°. The centre is at B. Outer 0 is on BA, so follow the outer scale to about 60°. The inner numbers at that position run the other way.
Now the starting arm slopes down the page. A horizontal baseline would not fit it. Turn the protractor with the arm and decide which of the two zeros belongs to that arm. Reading a sensible-looking number does not repair a misplaced centre or baseline.
Position the protractor and measure the smaller opening ∠ABC. This activity has no score. You can use paper and a real protractor, or the canvas tool.
Coordinates increase to the right and down the page. Lengths use equal coordinate scales.
| Object | Values | Given or drawn |
|---|---|---|
| Given arm BA | (100, 100) → (176, 138) | Given |
| Given arm BC | (100, 100) → (25, 170) | Given |
| B | (100, 100), included | Given |
| A | (176, 138), included | Given |
| C | (25, 170), included | Given |
The check looks for the centre at B, a baseline along an arm, both arms under the semicircle, and a whole-degree reading within 1°. This does not certify physical instrument use. You can also draw on paper and compare with the public answer.
Show solution
Coordinates increase to the right and down the page. Lengths use equal coordinate scales.
| Object | Values | Given or drawn |
|---|---|---|
| Given arm BA | (100, 100) → (176, 138) | Given |
| Given arm BC | (100, 100) → (25, 170) | Given |
| B | (100, 100), included | Given |
| A | (176, 138), included | Given |
| C | (25, 170), included | Given |
| Protractor | Protractor centre (100, 100); direction towards outer 0 (40, 70); radius 60. Small ticks are 1°. Labels are 30° apart. The outer scale increases from 0 to 180 and the inner scale decreases from 180 to 0. | Drawn |
About 110°. The zero on BA is on the inner scale. Follow it to about 110°, rather than taking the outer-scale reading near 70°. The centre is at B, and the sloping baseline follows BA.
Draw an angle of a given size
To draw 65°, first draw an arm from B to A. Put the protractor centre at B and its zero along BA. Mark the 65° position on the matching scale. Remove the protractor, then draw a straight arm from B through the mark. Label a point C on that arm.
Check with the protractor again. The angle is 65°; a longer BC does not make a larger angle.
Try that construction below. The given arm g begins at B. Draw your own second arm starting at B. Plan it with your protractor on paper if that helps, then place the arm on the canvas. A line elsewhere with the right slope would not make the requested angle at B.
Try the model. You can change your answer or look at the worked model. This activity has no score.
Coordinates increase to the right and down the page. Lengths use equal coordinate scales.
| Object | Values | Given or drawn |
|---|---|---|
| g | (40, 140) → (170, 140) | Given |
| B | (40, 140), included | Given |
The check looks for one straight new arm starting at B, making 65° with g within 1°. It inspects the drawing, not how you used a physical protractor. You can also draw on paper and compare with the public answer.
Show solution
Coordinates increase to the right and down the page. Lengths use equal coordinate scales.
| Object | Values | Given or drawn |
|---|---|---|
| g | (40, 140) → (170, 140) | Given |
| B | (40, 140), included | Given |
| example | (40, 140) → (90, 30); start unspecified, end unspecified | Drawn |
The digital check allows a small placement tolerance, stated with the task. Aim for the requested angle. Extending your arm along the same direction or entering its two endpoints in the other order keeps the opening unchanged.
Check your understanding 2 Reject a wrong reading
An angle is wider than a right angle. A protractor shows 55° on one scale and 125° on the other at its arm. Which is consistent with the opening? What must you check before accepting it?
Show answer
125° is wider than 90°. Check that the protractor centre is at the vertex, its baseline follows one arm and the chosen scale starts at zero on that arm. A sensible-looking number cannot repair a misplaced protractor.
Try a turned drawing
Draw 110° with the given arm sloping rather than horizontal. Turn the protractor so that its baseline follows g. Keep its centre at B. An angle measured from the edge of the screen would use the wrong starting direction.
Try the model. You can change your answer or look at the worked model. This activity has no score.
Coordinates increase to the right and down the page. Lengths use equal coordinate scales.
| Object | Values | Given or drawn |
|---|---|---|
| g | (100, 100) → (160, 130) | Given |
| B | (100, 100), included | Given |
The check looks for one straight new arm starting at B, making 110° with g within 1°. It inspects the drawing, not how you used a physical protractor. You can also draw on paper and compare with the public answer.
Show solution
Coordinates increase to the right and down the page. Lengths use equal coordinate scales.
| Object | Values | Given or drawn |
|---|---|---|
| g | (100, 100) → (160, 130) | Given |
| B | (100, 100), included | Given |
| example | (100, 100) → (25, 170); start unspecified, end unspecified | Drawn |
Explain how you aligned your protractor, then turn your paper drawing. The opening should keep its measure.
Show answer
The protractor’s baseline follows your chosen arm, however it slopes on the page. Its centre remains at the vertex. A rotation changes the drawing’s orientation, not the opening.
A typed degree value does not verify a drawing. The canvas check inspects your finished arm and opening. Practise the paper construction as well, and have someone inspect how you placed the ruler and protractor; the digital check does not observe that technique.
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