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End Of Chapter Exercises

Exercise 12.2See solutions

is a rhombus with and . Prove is also a rhombus.4061852e9abaf25b71bb23e25f868003.png

In and

(given rhombus , diagonal bisects )

(s opp equal sides)

similarly

and

but these are alternate interior s

and

is a parallelogram

(opp sides of m)

is a rhombus (all sides equal and two pairs of sides parallel)

is a parallelogram with diagonal . Given that , show that:

e695d05fb757a60e1b9cabd96f56ff24.png

is a parallelogram

Given parallelogram with bisecting and bisecting .b648dcfa077a4220c0ef2f0d984034c3.png

Write all interior angles in terms of .

First number the angles:e6d5479aa897cc613c6fffdd6e09d378.png

Prove that is a parallelogram.

Given that , and , prove that:a302882713cc4abd23cb521052df6a6e.png

bisects

First label the angles:3d763cfd48a98ce0d181b03f5a9063aa.png

is a point on , in . is the mid-point of . is the mid-point of and is the mid-point of . .273c36346c5740761973c1f4069cb5d1.png

Prove that is a parallelogram.

(given)

(given)

(Midpt Theorem)

(given)

(given)

(Midpt Theorem)

is a parallelogram (both pairs opp. sides parallel)

Prove that .

(given)

(given)

(Midpt Theorem)

(given)

In , , is the mid-point of and is the mid-point of .652b3227d8367824506122aff72f3596.png

Prove is the mid-point of .

If and the area of is , calculate the area of .

Prove that the area of will always be four times the area of , let and .

Given quadrilateral with sides and . Also given: and ; and . Complete the proof below to prove that is a parallelogram.2f3084927d748e5ddc89059d6f3e8eaa.png

The completed proof looks like this:

Calculate the value of .

is a parallelogram, .

Opposite s of parallelogram are equal. and .

Therefore, .

Calculate the value of .

s in a

Now we know that and that .

.

Study the quadrilateral with opposite angles and angles carefully. Fill in the correct reasons or steps to prove that the quadrilateral is a parallelogram.0153e365e117fbd29f0691acbcef84b2.png

Study the quadrilateral with and carefully. Fill in the correct reasons or steps to prove that the quadrilateral is a parallelogram.864e69615e3b8245a7aa078f80f4abe7.png

In parallelogram , the bisectors of the angles have been constructed, indicated with the red lines below. You are also given , , , , and .

Prove that the quadrilateral is a parallelogram.

Note the diagram is drawn to scale.5b9364c3147234964fcbe352215aa298.png

Redraw the diagram and mark all the known information:2ee5f8218f88ef99d1c4c5ccc33d9dd8.png

Study the diagram below; it is not necessarily drawn to scale. Two triangles in the figure are congruent: . Additionally, . You need to prove that is a parallelogram.8eeeabe4474dcabdef3dbd523b130a79.png

Redraw the diagram and mark all known and given information:312469f7e2fda370239a3ddbf60a54e7.png

Given the following diagram:db5c101c981e19588b128f911fda5baa.png

Show that is a parallelogram.

Show that is a parallelogram.

But and are diagonals of , is a parallelogram (diagonals bisect each other).

Prove that .

(proved above).

is a parallelogram. is a parallelogram. is a straight line. Prove that .0116158dd5bb5d6d97a9ce0ede75fbf9.png

In the figure below , . is a parallelogram. Prove is a straight line.272898e6e1776d61a3d380aafa9647ad.png

We note that:

We also note that:

Now we can show that is congruent to :

Finally we can show that is a straight line:

See solutions