3.7 Midsegments of Triangles

Notes Midsegment of a Triangle: a segment whose endpoints are the midpoints of two sides.

Activity 1: Triangle Midsegments Draw Δ ABC. Find the midpoints, M and N, of sides AB and AC. Then draw MN, the midsegment. Measure and record MN and BC on your paper. What is the relationship between their lengths? Measure and . Measure and . What do your measurements suggest about BC and MN? What postulate or theorem allows you to draw this conclusion? Rewrite and complete the following conjecture: Triangle Midsegment Conjecture A midsegment of a triangle is _______________ to a side of the triangle and has a measure equal to ___________________ of that side. 2 1 3 4 C B

Triangle Congruences

What does it mean for two figures to be congruent? They must have the same: SIZE SHAPE

Here’s the situation… Prior to the start of a sailboat race, you (the judging official) must certify that all of the sails are the same size. Without unrigging the triangular sails from their masts, how can the official (you) determine if the sails on each of the boats are the same size? With your group discuss and write down how you would go about doing this? Over the next couple of classes we will be learning some geometry tricks (concepts) involving triangles that will help us answer the above question. Hand out materials

Activity 2: SSS Postulate Using these three objects, create a triangle. (The three sides being the ruler, unsharpened pencil and straightedge of the protractor.) Compare your triangle with your group members triangles. What do you notice? Did everyone create the same triangle? Are all of your triangles congruent? Yes Why? All of the parts are the same or congruent. Notice that we did not even pay any attention to the angles and they “took care of themselves” Create another triangle using the three objects, but this time only using 8 inches of the ruler for one of the sides. Are all of your triangles congruent again? With your group discuss how we can use this concept to relate back to our initial problem with the sailboats.

SSS (Side-Side-side) Postulate If the sides of one triangle are congruent to the sides of another triangle, then the two triangles are congruent.

Activity 3: SAS Postulate Draw a 6 cm segment. Label it GH. Using your protractor, make  G = 60. From vertex G, draw GI measuring 7 cm long. Label the end point I. From the given information, how many different triangles can be formed? Form  GHI. Is your  GHI congruent to your group members  GHI. What information was used to create this triangle? Draw another segment this time 10 cm long. Label it XY. Using your protractor, make  X = 45. From vertex X, draw XZ measuring 5 cm long. Label the end point Z. How many different triangles can be formed? Form  XYZ. Is your  XYZ congruent to your group members  XYZ?

SAS (Side-Angle-side) Postulate If two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle, then the two triangles are congruent.

Triangle Congruences

ASA (Angle-Side-Angle) Postulate If two angles and the included side in one triangle are congruent to two angles and the included side in another triangle, then the two triangles are congruent.

Practice In each pair below, the triangles are congruent. Tell which triangle congruence postulate allows you to conclude that they are congruent, based on the markings in the figures.

AAS (Angle-Angle-Side) Postulate If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent.

Practice Which pairs of triangles below can be proven to be congruent by the AAS Congruence Theorem?

Two other possibilities AAA combination—three angles Does it work? SSA combination—two sides and an angle that is not between them (that is, an angle opposite one of the two sides.)

Special case of SSA When you try to draw a triangle for an SSA combination, the side opposite the given angle can sometimes pivot like a swinging door between two possible positions. This “swinging door” effect shows that two triangles are possible for certain SSA information.

A special case of ssa If the given angle is a right angle, SSA can be used to prove congruence. In this case, it is called the Hypotenuse-Leg Congruence Theorem.

Review of Right Triangles

HL (Hypotenuse-Leg) Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the Hypotenuse and a leg of another right triangle, then the two triangles are congruent.

Other Right Triangle Theorems LL (LEG-LEG) Congruence Theorem If the two legs of a right triangle are congruent to the corresponding two legs of another right triangle, then the triangles are congruent. LA (LEG-ANGLE) Congruence Theorem If a leg and an acute angle of a right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the triangles are congruent. Right Triangle version of ______ Right Triangle version of ______

Other Right Triangle Theorems HA (HYPOTENUSE-ANGLE) Congruence Theorem If the hypotenuse and an acute angle of a right triangle are congruent to the corresponding hypotenuse and acute angle of another triangle, then the triangles are congruent. HL (HYPOTENUSE-LEG) Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the corresponding hypotenuse and leg of another right triangle, then the triangles are congruent. Right Triangle version of ______