![]() ![]() ![]() The same thing goes wrong in this case but it is interesting to consider and provides an opportunity to study some of the special types of parallelograms. We can look at what happens in the special case where all 4 sides of both $ABCD$ and $EFGH$ are congruent to one another. The same is true of parallelogram $EFGH$ (which is obtained by adjoining $\triangle GHE$ to $\triangle EFG$) and since $\triangle ABC$ is congruent to $\triangle EFG$ (and $\triangle CDA$ is congruent to $\triangle GHE$) we can conclude that parallelogram $ABCD$ is congruent to parallelogram $EFGH$.Ģ Looking at a special case for part (a): the rhombus In other words the parallelogram $ABCD$ is obtained by adjoining to $\triangle ABC$ a second triangle, $\triangle CDA$, which is congruent to This is pictured below with the image of $B$ labeled $D$: The opposite sides of a parallelogram are congruent so we will need two pairs of congruent segments: We begin by drawing or building a parallelogram. Thus it provides a good opportunity for students to engage in MP3 ''Construct Viable Arguments and Critique the Reasoning of Others.'' Also as noted above, students working on this task have multiple opportunities to engage in MP5 ''Use Appropriate Tools Strategically'' as they can use manipulatives or computer software to experiment with constructing different parallelograms. This task would be ideally suited for group work since it is open ended and calls for experimentation. For example, for squares one side is enough, for rectangles two adjacent sides are sufficient. Four sides is not enough, but what about other combinations such as SASA? An interesting extension of this activity would be to have students make and verify conjectures about how much information is needed to determine if two quadrilaterals are congruent. Just as with a triangle it takes three pieces of information (ASA, SAS, or SSS) to determine a shape, so with a quadrilateral we would expect to require four pieces of information. For quadrilaterals, on the other hand, four toothpicks can be put together to make any of the rhombuses with that side length. If manipulatives are available, it would be valuable to use toothpicks for example to see that with three of them only one triangular shape is possible, namely an equilateral triangle. Unlike with triangles, some information about angles is needed in order to conclude that two quadrilaterals are congruent. It turns out that knowing all four sides of two quadrilaterals are congruent is not enough to conclude that the quadrilaterals are congruent. This task is ideal for hands-on work or work with a computer to help visualize the possibilities. This task addresses this issue for a specific class of quadrilaterals, namely parallelograms. For quadrilaterals, on the other hand, these nice tests seem to be lacking. RHS rule states that if in a right angled triangle hypotenuse and one side are equal, the two triangles are congruent.Triangle congruence criteria have been part of the geometry curriculum for centuries.ASA congruence rule states that if two angles and a side in the middle of the two angle are equal, the triangles are congruent.SAS congruence rule states that if two sides and an angle in the middle of the two sides are equal, the two triangles are congruent.SSS congruence rule states that if all sides of a triangle are equal, the triangles are congruent.There are 4 rules to determine if two triangles are congruent: SSS, SAS, ASA, RHS.Congruent triangles are those triangles whose sides and angles are exactly equal.If all the three corresponding sides of two triangles are equal then they are said to be congruent by SSS rule. RHS (Right angle- Hypotenuse-Side)- If the hypotenuse and side of one right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, the two triangles are congruent.AAS(Angle-Angle-Side)- Two triangles are said to be congruent by AAS condition if their two angles and 1 side are equal.ASA(Angle-Side-Angle)- Two triangles are said to be congruent by ASA condition if their two angles and 1 side are equal. ![]() SAS(Side-Angle-Side)- Two triangles are said to be congruent by SAS condition if their two sides and 1 angle are equal.SSS (Side-Side-Side) – Two triangles are said to be congruent by SSS condition if all three sides are equal.If there are two triangles A and B then if they fulfil any of the below mentioned conditions then they are said to be congruent and they are mentioned like below: So, we can say both triangles ABC and PQR are congruent. ![]()
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