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constructing congruent triangles
Transcript of constructing congruent triangles
GIVEN & PROVEN
ASA and AAS Theorems
Sean Anthony Martinez
Remember the theorems based off of the sides and angles
Make sure to use a compass to find and write points
Don't forget that AAA and SSA does not exist!!!
I WILL BE
IN CHAPT. 4
"Boxes aren't necessarily bad places to think in."
WHY SSA AND AAA WILL NOT WORK
SSS and SAS Postulates
Why AAA and SSA Will not Work
3 most Important notes to know
Step 1: Draw a line segment with the Given length with the straight edge
The Given problem example would be to "construct an angle with the sides 5 inches, 6 inches, and 3 inches
Step 2: Set the width of your compass equal to another given side length
Step 3: Place the tip of the compass on one of the end points of the side AB (let's choose to place it on A) and draw an arc on either side of the line segment AB.
Step 4: Set the width of your compass equal to length of the third side, which is 4cm in this case.
Step 5: Place the tip of the compass on B and draw another arc which cuts the previously drawn arc at some point (say C).
Step 6: Join C to each of the points A and B to complete the triangle.
If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent
If... AB=DE, BC=EF
Step 1: Construct an angle of the given measure (50o) in this case. Let's name the vertex of this angle as point A. Ensure that each of the arms of this angle are longer than the given two side lengths.
Step 2: Set the width of your compass equal to the one of the given side lengths. Let's set it equal to 5cm. Retain this width for the next step.
Step 3: Place the tip of the compass on the vertex of the 50o angle (point A) and draw an arc so as to cut any one of its arms at some point (say B).
Step 1: Start with the given line segment and two angles. Mark a point A that will be one vertex of the new triangle.
Step 2: Set the compasses' width to the length of the segment AB.
Step 4: Mark a point B on this arc. Then draw the line AB. This will be one side of the new triangle.
Step 5: With the compasses at any convenient width, draw an arc across both lines of the given angle A.
Step 6: Without changing the compasses' width, draw an arc at point A on the new triangle. The arc must cross AB
Step 7: Set the compasses to the arc width at the given angle A. This the distance between the points where the arc intersects the sides of the angle.
If two sides and the included side of one triangle are congruent to two angles an the included side of another triangle, then the two triangles are congruent
If A=D, AC=DF,C=F
If two angles and a non-include side of one triangle are congruent to two angles and the corresponding non included side of another triangle, then the triangles are congruent
Uses the third angles theorem
AAA will not work because it can only make similar, but not congruent triangles
The angles are corresponding. but it cannot be truly true with the sides
They can then be any size
SSA will not work because the unknown side could be located in two different places and we can create two different triangles
Can only be true with right angles
Step 4: Set the width of your compass equal to the other given side length, which would be 7cm in this case.
Step 5: Place the tip of the compass on the vertex of the 50o angle (point A) and draw an arc so as to cut the other arm at some point (say C).
Step 6: Connect the points B and C.
If the hypotenuse and a leg of one right triangle are are congruent to the hypotenuse an a leg of another right triangle, then the triangles are congruent
Step 1: Use the Pythagorean theorem or make sure there is a hypotenuse and a leg.
Step 2: make sure there is the angle-side-side angle with a right angle
Step 3: Do what other information is needed. In this case, what is needed to be known is CB=YZ
SIDE-ANGLE SIDE Postulate
If two sides and the included side of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent
If... A=D, B=E, AC=DF
WHAT IS TRIANGLE CONGRUENCE?
If two triangles are congruent they will have exactly the same three sides and exactly the same three angles.
Step 3: With the compasses' point on A, make an arc near the future vertex B of the triangle.
Step 8: Near point A draw an arc in a similar position so it crosses the arc drawn earlier. This, in effect, 'copies' the measure of the angle at P to the angle at A.
Step 9: Draw a line with a straight edge from A through the point where the arcs intersect. This will become the second side of the triangle. Draw it long.
Step 10: Repeat this process at B. Copying the angle measure from the given angle B to the new triangle at B. The point where the lines intersect is C, the third vertex of the triangle