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Geometry Problem URGENTTTTTT

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Geometry Problem URGENTTTTTT

by arifaisal » Thu Nov 10, 2011 8:05 am
Q1 .In this diagram, AB=AC. and BD=CD. Which of the following statements is true?
A. BE=EC
B. AD is perpendicular to BC
C. triangle BDE and CDE are congruent
D. angle ABD equal angle ACD
E. All of these

Figure: SEE ATTACHMENT

Please give explanation for each of these why something is not true

Q2. ABCD is a rectangle; the diagonals AC and BD intersect at E. Which of the following statements is not necessarily true?

A. AE=BE
B. Angle AEB equals angle CED
C. AE is perpendicular to BD
D. Triangles AED and AEB are equal in area
E. Angle BAC equals angle BDC

Pls give full explanation
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by GmatMathPro » Thu Nov 10, 2011 8:54 am
I HOPE THIS IS IN TIME!!!!!!!!!!!!
arifaisal wrote:Q1 .In this diagram, AB=AC. and BD=CD. Which of the following statements is true?
A. BE=EC
B. AD is perpendicular to BC
C. triangle BDE and CDE are congruent
D. angle ABD equal angle ACD
E. All of these
If AB=AC and BD=CD, then triangle ABD and triangle ACD are congruent by SSS postulate. Therefore, angles BAE and CAE are congruent. Now, we know that AB=AC(given), AE=AE(reflexive), and angle BAE=angle CAE(corresponding parts of congruent triangles are congruent), so triangles BAE and CAE are congruent by SAS postulate. Thus, angle BEA and angle CEA are congruent. But they also are supplementary (add to 180), so they must both be right angles. Also, BE=CE because they are corresponding parts of congruent triangles. So, in sum, AE is perpendicular to BC AND bisects BC, so AD is the perpendicular bisector of BC.

A. BE=EC is true as shown above
B. AD is perpendicular to BC is true as shown above
C. triangle BDE and CDE are congruent: BD=CD, DE=DE, BE=CE, therefore they are congruent by SSS. TRUE.
D. angle ABD=angle ACD is true because triangle ABD is congruent to triangle ACD as shown above.
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by GmatMathPro » Thu Nov 10, 2011 9:06 am
arifaisal wrote: Q2. ABCD is a rectangle; the diagonals AC and BD intersect at E. Which of the following statements is not necessarily true?

A. AE=BE
B. Angle AEB equals angle CED
C. AE is perpendicular to BD
D. Triangles AED and AEB are equal in area
E. Angle BAC equals angle BDC

Pls give full explanation
A. AE=BE is always true. All rectangles are parallelograms and all parallelograms have diagonals that bisect each other. AC=BD because all rectangles have congruent diagonals. Thus the halves of two congruent things will also be congruent.
B. angle AEB is always equal to angle CED because they are vertical angles.
C. AE will only be perpendicular to BD if this rectangle is also a square.
D. triangles AED and AEB are always equal in area. Consider right triangle DAB with base BD. AE bisects BD, so triangles AED and AEB will each have bases DE and EB respectively that are congruent. They will also both share the altitude drawn from A. With congruent bases and heights these triangles will always have equal areas.
E. Angle BAC always equals angle BDC: Triangle DEC is isosceles, therefore angle EDC is congruent to angle ECD. AB is parallel to CD, so angle BAC is congruent to angle ECD because they are alternate interior angles. So, angle BAC=angle ECD=angle EDC, and EDC is the same as BDC, so angle BAC=angle BDC.
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by arifaisal » Thu Nov 10, 2011 9:51 am
thanks man...you are awesome
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