Exploring congruent triangles , math homework help

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Exploring congruent triangles , math homework help

Congruent Triangles

You have learned five ways to prove that triangles are congruent: SSS, SAS, ASA, AAS, and HL. You are going to model these theorems with string and a protractor.

Example: Recreate the following triangle using HL.

    A right triangle is shown. One side is vertical, one side is horizontal, and one side is diagonal. The right angle is between the vertical and horizontal sides.

  1. HL requires a right angle, so use a protractor to create a right angle.

    A right triangle is shown with vertical and horizontal sides.

  2. Choose a leg of the original right triangle. Cut a piece of string the length of that leg.

    A right triangle is shown. One side is vertical, one side is horizontal, and one side is diagonal. The right angle is between the vertical and horizontal sides. A line segment of the same length as the vertical side is next to the vertical side.

  3. Lay the string along the corresponding leg of the right angle that you drew in Step 1 and make the leg the same length as the string by either extending the side or erasing part of the side.

    A right angle is shown with vertical and horizontal sides. A line segment is next to the vertical side. The vertical side is longer than the line segment and so the top of the vertical side is crossed out to make them the same length.

  4. Cut a piece of string the length of the hypotenuse of the original right triangle.

    A right triangle is shown. One side is vertical, one side is horizontal and one side is diagonal. The right angle is between the vertical and horizontal sides. A line segment of the same length as the diagonal side is next to the diagonal side.

  5. Lay one end of the string at the top of the leg that you created in Step 3. Make the other end of the string intersect the other side of the right angle, forming a triangle. Trace the string. Make the bottom leg of the triangle the correct length by either extending the side or erasing part of the side.

  6. You’ve now copied the triangle using HL.

    The figure shows a right triangle. One side is vertical, one side is horizontal and one side is diagonal. The right angle is between the vertical and horizontal sides.

    Proving Triangles Congruent

    Complete the following table to summarize the different ways to prove triangles congruent.

    Postulate/Theorem What It Says Required Information Picture
    SSS If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent. three sets of congruent sides Two triangles have three sets of congruent sides marked.
    SAS
    ASA
    AAS
    HL

    You will turn in the chart above.

    Recreating the Original Triangle

    You will be using the following original triangle in the following activity.

    With a ruler, measure the sides of the original triangle to the nearest millimeter. Write down the length of each side in your math journal.With a protractor, measure the angles of the original triangle to the nearest degree. Write down the measure of each angle in your math journal.

    SSS: Cut three pieces of string. Make each piece of string the length of one of the sides of the original triangle. Put the string together to form a triangle and trace the triangle on a separate piece of paper. Measure the angles of the triangle with your protractor.Answer the following questions in your math journal:

    1. Are the lengths of the sides and the measures of the angles of the triangle you created the same as the original triangle?
    2. Rearrange the string to make a different triangle. Is there any way to create a triangle that has different angle measures?

    SAS: Choose two sides of the original triangle. Cut two pieces of string and make each piece of string the length of one of those sides. Measure out the angles at both endpoints of the side that you chose. Draw the angles with the given measurements. Put the string together to form the sides of that angle and trace them. Draw in the third side of the triangle. Measure the third side that you drew and the two angles adjacent to that side.Answer the following questions in your math journal:

    1. Are the lengths of the sides and the measures of the angles of the triangle you created the same as the original triangle?
    2. Draw the starting angle elsewhere on your paper and rearrange the string to make a different triangle. Is there any way to create a triangle whose third side has a different length?

    ASA: Choose one side of the original triangle. Cut one piece of string and make the piece of string the length of that side. Trace the string on a separate sheet of paper. Measure out the angles at both endpoints of the side that you chose. Draw the angles with the given measurements. Extend the sides of the angles until they intersect and form a triangle. Measure the two sides that you drew and the angle between them.Answer the following questions in your math journal:

    1. Are the lengths of the sides and the measures of the angles of the triangle you created the same as the original triangle?
    2. Rearrange the string and re-draw the two starting angles to make a different triangle. Is there any way to create a triangle that has different side lengths?

    You will need to submit the following:

    • the answers to questions 1–6 above
    • your re-drawn triangles with the pieces of string that you cut taped to each drawing

      You will need to submit the following:

      • the steps that you would follow to recreate a triangle using a protractor, a string, and the AAS Congruence Theorem, written in your own words

      At the end of the assignment, you will need to submit the following:

      • a chart summarizing all of the postulates and theorems that can be used to prove two triangles congruent
      • the answers to six questions about recreating a triangle using a protractor, string, and the SSS, SAS, and ASA Congruence Postulates
      • three drawings of recreated triangles, with the string used to create the drawings taped to the appropriate drawing
      • a description of the steps that you would follow to recreate a triangle using a protractor, a string, and the AAS Congruence Theorem, written in your own words