How to Construct a Triangle when Two Sides and the Included Angle are Given (SAS Criterion)
Set your compass to the length of a chosen side., Copy the length., Connect the points., Choose any compass setting., Draw arcs around point A., Copy the arc onto the new triangle construction., Set the compass to the angle width., Copy the angle...
Step-by-Step Guide
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Step 1: Set your compass to the length of a chosen side.
Begin with a chosen triangle with points labeled A{\displaystyle A}, B{\displaystyle B}, and C{\displaystyle C}.
Set one point of your compass on point A{\displaystyle A}, and open your compass so the pencil is on point B{\displaystyle B}.
Fix your compass so this measurement is set.Some compasses are more flexible than others.
When you set this length, you need to be sure that the compass does not move as you transfer to your new figure.
There may be a screw to tighten or a clip to adjust to set this length. -
Step 2: Copy the length.
At a new position on your paper, wherever you wish to draw the new triangle, mark a point and label it A1{\displaystyle A_{1}}.
Set the point of your compass on A1{\displaystyle A_{1}}.
Mark a point with the compass pencil at the designated length.
Label this point B1{\displaystyle B_{1}}., Use your straightedge to connect points A1{\displaystyle A_{1}} and B1{\displaystyle B_{1}}.
This line should be an exact copy of side AB¯{\displaystyle {\overline {AB}}} from the original triangle.
If segment A1B1¯{\displaystyle {\overline {A_{1}B_{1}}}} does not look like it is the same length as original side AB¯{\displaystyle {\overline {AB}}}, then your compass probably slipped while you were moving it.
Start over and try again. , To begin copying the angle, set your compass to any reasonable size.
Choose a setting that is roughly half the length of the sides of your original triangle. , Set the point of your compass on point A{\displaystyle A} of the original triangle.
Swing the compass so it draws an arc that intersects both sides AB¯{\displaystyle {\overline {AB}}} and AC¯{\displaystyle {\overline {AC}}}.Mark the two positions where the arc intersects the triangle sides.
Label the intersection with side AB¯{\displaystyle {\overline {AB}}} as x{\displaystyle x} and label the intersection with side AC¯{\displaystyle {\overline {AC}}} as y{\displaystyle y}. , Without letting the compass setting change, place the point of the compass on A1{\displaystyle A_{1}} of your construction.
Draw an arc that intersects side A1B1¯{\displaystyle {\overline {A_{1}B_{1}}}} and continues about as long as the arc you drew on the original triangle.
Mark the position where the arc intersects side A1B1¯{\displaystyle {\overline {A_{1}B_{1}}}} as x1{\displaystyle x_{1}}. , Pick up the compass and set the point on point x{\displaystyle x} on the original triangle.
Set the width of the compass by placing the pencil tip on y{\displaystyle y}., Position your compass so the point is on x1{\displaystyle x_{1}}.
Draw a small arc that intersects the arc that you previously drew.
Mark the point where these two arcs intersect as y1{\displaystyle y_{1}}., Using your straightedge, draw a line that connects points A1{\displaystyle A_{1}} and y1{\displaystyle y_{1}}.
Extend the line beyond y1{\displaystyle y_{1}} so it will be slightly longer than side AB¯{\displaystyle {\overline {AB}}} of your original triangle.
This angle, x1Ay1{\displaystyle x_{1}Ay_{1}} should be an exact duplicate of angle BAC{\displaystyle BAC} of the original triangle.
If it doesn’t look the same, you should repeat the process and make sure that your compass does not slip during the transfers. , Place the tip of the compass on point A{\displaystyle A}.
Place the pencil tip on point C{\displaystyle C}.
Set the length of the compass so it does not shift. , Move the tip of the compass to point A1{\displaystyle A_{1}} of the construction.
Using the compass pencil, draw a small arc that intersects the extended line A1y1{\displaystyle A_{1}y_{1}} that you previously constructed.
Mark the point of intersection as C1{\displaystyle C_{1}}.The length of A1C1¯{\displaystyle {\overline {A_{1}C_{1}}}} should be the same as the length of A1C1¯{\displaystyle {\overline {A1C1}}}.
If not, then try again. , Using your straightedge, connect points B1{\displaystyle B_{1}} and C1{\displaystyle C_{1}} on your construction.
This should finish triangle A1B1C1{\displaystyle A_{1}B_{1}C_{1}} as an exact copy of triangle ABC{\displaystyle ABC}.If you wish, you can clean up your construction by erasing the arc lines that you drew for the construction. , You can also construct a duplicate triangle, using two given sides and the angle between them (SAS), using a ruler and protractor.
This is not the true classical method, but it is easier if you are allowed to use these extra tools.
Use your ruler to measure the length of side AB{\displaystyle AB}., At a new position on your paper, where you want to draw your new triangle, mark a point A1{\displaystyle A_{1}}.
Set your ruler at point A1{\displaystyle A_{1}} and draw a line segment that is the same length as side AB{\displaystyle AB}.
Label the end of this line segment B1{\displaystyle B_{1}}. , On the original triangle, measure angle BAC{\displaystyle BAC} with your protractor.
Lift the protractor and set the center point on point A1{\displaystyle A_{1}} of the new construction.
Mark an angle the same size as angle BAC{\displaystyle BAC}.
Use your straightedge to connect A1{\displaystyle A_{1}} and this mark.
Extend the line slightly longer than the length of side AC¯{\displaystyle {\overline {AC}}} of the original triangle. , Using the ruler, measure the length of side AC{\displaystyle AC} of the original triangle. , Place one end of the ruler at point A1{\displaystyle A_{1}}.
Along the line that you drew in the previous step, mark a length equal to the measured length of side AC¯{\displaystyle {\overline {AC}}}.
Label the end of this length C1{\displaystyle C_{1}}., Connect points B1{\displaystyle B_{1}} and C1</math.Thisshouldcompletetheconstructionoftriangle<math>A1B1C1{\displaystyle C_{1}</math.Thisshouldcompletetheconstructionoftriangle<math>A_{1}B_{1}C_{1}} as a congruent copy of triangle ABC{\displaystyle ABC}.If you wish, you can erase the markings that you made during the construction. -
Step 3: Connect the points.
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Step 4: Choose any compass setting.
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Step 5: Draw arcs around point A.
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Step 6: Copy the arc onto the new triangle construction.
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Step 7: Set the compass to the angle width.
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Step 8: Copy the angle width to the new construction.
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Step 9: Complete the angle.
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Step 10: Set your compass to the length of AC.
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Step 11: Transfer the length to your construction.
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Step 12: Connect points B1{\displaystyle B_{1}} and C1{\displaystyle C_{1}}.
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Step 13: Measure side AB{\displaystyle AB}.
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Step 14: Copy the side length to the new construction.
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Step 15: Copy angle BAC.
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Step 16: Measure side AC{\displaystyle AC}.
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Step 17: Copy the side length to the new construction.
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Step 18: Complete the triangle.
Detailed Guide
Begin with a chosen triangle with points labeled A{\displaystyle A}, B{\displaystyle B}, and C{\displaystyle C}.
Set one point of your compass on point A{\displaystyle A}, and open your compass so the pencil is on point B{\displaystyle B}.
Fix your compass so this measurement is set.Some compasses are more flexible than others.
When you set this length, you need to be sure that the compass does not move as you transfer to your new figure.
There may be a screw to tighten or a clip to adjust to set this length.
At a new position on your paper, wherever you wish to draw the new triangle, mark a point and label it A1{\displaystyle A_{1}}.
Set the point of your compass on A1{\displaystyle A_{1}}.
Mark a point with the compass pencil at the designated length.
Label this point B1{\displaystyle B_{1}}., Use your straightedge to connect points A1{\displaystyle A_{1}} and B1{\displaystyle B_{1}}.
This line should be an exact copy of side AB¯{\displaystyle {\overline {AB}}} from the original triangle.
If segment A1B1¯{\displaystyle {\overline {A_{1}B_{1}}}} does not look like it is the same length as original side AB¯{\displaystyle {\overline {AB}}}, then your compass probably slipped while you were moving it.
Start over and try again. , To begin copying the angle, set your compass to any reasonable size.
Choose a setting that is roughly half the length of the sides of your original triangle. , Set the point of your compass on point A{\displaystyle A} of the original triangle.
Swing the compass so it draws an arc that intersects both sides AB¯{\displaystyle {\overline {AB}}} and AC¯{\displaystyle {\overline {AC}}}.Mark the two positions where the arc intersects the triangle sides.
Label the intersection with side AB¯{\displaystyle {\overline {AB}}} as x{\displaystyle x} and label the intersection with side AC¯{\displaystyle {\overline {AC}}} as y{\displaystyle y}. , Without letting the compass setting change, place the point of the compass on A1{\displaystyle A_{1}} of your construction.
Draw an arc that intersects side A1B1¯{\displaystyle {\overline {A_{1}B_{1}}}} and continues about as long as the arc you drew on the original triangle.
Mark the position where the arc intersects side A1B1¯{\displaystyle {\overline {A_{1}B_{1}}}} as x1{\displaystyle x_{1}}. , Pick up the compass and set the point on point x{\displaystyle x} on the original triangle.
Set the width of the compass by placing the pencil tip on y{\displaystyle y}., Position your compass so the point is on x1{\displaystyle x_{1}}.
Draw a small arc that intersects the arc that you previously drew.
Mark the point where these two arcs intersect as y1{\displaystyle y_{1}}., Using your straightedge, draw a line that connects points A1{\displaystyle A_{1}} and y1{\displaystyle y_{1}}.
Extend the line beyond y1{\displaystyle y_{1}} so it will be slightly longer than side AB¯{\displaystyle {\overline {AB}}} of your original triangle.
This angle, x1Ay1{\displaystyle x_{1}Ay_{1}} should be an exact duplicate of angle BAC{\displaystyle BAC} of the original triangle.
If it doesn’t look the same, you should repeat the process and make sure that your compass does not slip during the transfers. , Place the tip of the compass on point A{\displaystyle A}.
Place the pencil tip on point C{\displaystyle C}.
Set the length of the compass so it does not shift. , Move the tip of the compass to point A1{\displaystyle A_{1}} of the construction.
Using the compass pencil, draw a small arc that intersects the extended line A1y1{\displaystyle A_{1}y_{1}} that you previously constructed.
Mark the point of intersection as C1{\displaystyle C_{1}}.The length of A1C1¯{\displaystyle {\overline {A_{1}C_{1}}}} should be the same as the length of A1C1¯{\displaystyle {\overline {A1C1}}}.
If not, then try again. , Using your straightedge, connect points B1{\displaystyle B_{1}} and C1{\displaystyle C_{1}} on your construction.
This should finish triangle A1B1C1{\displaystyle A_{1}B_{1}C_{1}} as an exact copy of triangle ABC{\displaystyle ABC}.If you wish, you can clean up your construction by erasing the arc lines that you drew for the construction. , You can also construct a duplicate triangle, using two given sides and the angle between them (SAS), using a ruler and protractor.
This is not the true classical method, but it is easier if you are allowed to use these extra tools.
Use your ruler to measure the length of side AB{\displaystyle AB}., At a new position on your paper, where you want to draw your new triangle, mark a point A1{\displaystyle A_{1}}.
Set your ruler at point A1{\displaystyle A_{1}} and draw a line segment that is the same length as side AB{\displaystyle AB}.
Label the end of this line segment B1{\displaystyle B_{1}}. , On the original triangle, measure angle BAC{\displaystyle BAC} with your protractor.
Lift the protractor and set the center point on point A1{\displaystyle A_{1}} of the new construction.
Mark an angle the same size as angle BAC{\displaystyle BAC}.
Use your straightedge to connect A1{\displaystyle A_{1}} and this mark.
Extend the line slightly longer than the length of side AC¯{\displaystyle {\overline {AC}}} of the original triangle. , Using the ruler, measure the length of side AC{\displaystyle AC} of the original triangle. , Place one end of the ruler at point A1{\displaystyle A_{1}}.
Along the line that you drew in the previous step, mark a length equal to the measured length of side AC¯{\displaystyle {\overline {AC}}}.
Label the end of this length C1{\displaystyle C_{1}}., Connect points B1{\displaystyle B_{1}} and C1</math.Thisshouldcompletetheconstructionoftriangle<math>A1B1C1{\displaystyle C_{1}</math.Thisshouldcompletetheconstructionoftriangle<math>A_{1}B_{1}C_{1}} as a congruent copy of triangle ABC{\displaystyle ABC}.If you wish, you can erase the markings that you made during the construction.
About the Author
Amy Edwards
Enthusiastic about teaching practical skills techniques through clear, step-by-step guides.
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