# Thread: Help with these two geometry!

1. ## Help with these two geometry!

Help with these two hard problems:

Problem 12
Let M be a point inside the angle A. From point M perpendiculars MP and MQ are dropped to the angle sides. From point A, perpendicular AK is dropped to segment PQ. Prove that angle PAK = angle MAQ.
Problem 13
Let ABC be a triangle and angle B = 60. Angle bisectors AD and CE meet at the point O. Prove that OD = OE.

2. Hello xxravenxx
Originally Posted by xxravenxx
... Problem 12
Let M be a point inside the angle A. From point M perpendiculars MP and MQ are dropped to the angle sides. From point A, perpendicular AK is dropped to segment PQ. Prove that angle PAK = angle MAQ.
$\angle APM = \angle AQM = 90^o$ (given)

$\Rightarrow APMQ$ is a cyclic quadrilateral (opp angles supplementary)

$\Rightarrow \angle APQ = \angle AMQ$ (angles in same segment)

So in $\triangle$'s $APK, AMQ:$
$\angle APK = \angle AMQ$ (proven)
$\angle AKP = \angle AQM$ ( $=90^o$, given)
$\Rightarrow \angle PAK = \angle MAQ$ (angle sum of triangle)

3. Sorry but can you draw me a diagram of what this looks like? I'm a bit clueless.

4. Hello xxravenxx
Originally Posted by xxravenxx
Sorry but can you draw me a diagram of what this looks like? I'm a bit clueless.
See attachment.

... and here's the answer for number 13:
Originally Posted by xxravenxx
...Problem 13
Let ABC be a triangle and angle B = 60. Angle bisectors AD and CE meet at the point O. Prove that OD = OE.
(I hope you can draw a diagram for this.)

Let $\angle BAD = \angle DAC = x^o$ and $\angle ACE = \angle ECB = y^o$

Then, $2x+2y+ 60=180$ (angle sum of $\triangle ABC$)

$\Rightarrow x+y=60$

Now $\angle DOC$ is an exterior angle of $\triangle OAC$ = sum of interior opposite angles $= x+y = 60^o$

$\Rightarrow BEOD$ is a cyclic quadrilateral (exterior angle $\angle DOC$ = interior opposite angle $\angle B$)

Now the angle bisectors of a triangle are concurrent $\Rightarrow BO$ is the bisector of $\angle ABC$

$\Rightarrow \angle DBO = \angle EBO$

$\Rightarrow \angle DEO = \angle EDO$ (angles in same segment)

$\Rightarrow OE = OD$ (isosceles $\triangle OED$)
The text contains two geometry problems, each with a detailed solution provided. The first problem involves proving the equality of two angles, and the second problem involves proving the equality of two segments in a triangle. Diagrams are requested and provided for the first problem, and the solution for the second problem is also given.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Thread: Help with these two geometry!

1. ## Help with these two geometry!

Help with these two hard problems:

Problem 12
Let M be a point inside the angle A. From point M perpendiculars MP and MQ are dropped to the angle sides. From point A, perpendicular AK is dropped to segment PQ. Prove that angle PAK = angle MAQ.
Problem 13
Let ABC be a triangle and angle B = 60. Angle bisectors AD and CE meet at the point O. Prove that OD = OE.

2. Hello xxravenxx
Originally Posted by xxravenxx
... Problem 12
Let M be a point inside the angle A. From point M perpendiculars MP and MQ are dropped to the angle sides. From point A, perpendicular AK is dropped to segment PQ. Prove that angle PAK = angle MAQ.
$\angle APM = \angle AQM = 90^o$ (given)

$\Rightarrow APMQ$ is a cyclic quadrilateral (opp angles supplementary)

$\Rightarrow \angle APQ = \angle AMQ$ (angles in same segment)

So in $\triangle$'s $APK, AMQ:$
$\angle APK = \angle AMQ$ (proven)
$\angle AKP = \angle AQM$ ( $=90^o$, given)
$\Rightarrow \angle PAK = \angle MAQ$ (angle sum of triangle)

3. Sorry but can you draw me a diagram of what this looks like? I'm a bit clueless.

4. Hello xxravenxx
Originally Posted by xxravenxx
Sorry but can you draw me a diagram of what this looks like? I'm a bit clueless.
See attachment.

... and here's the answer for number 13:
Originally Posted by xxravenxx
...Problem 13
Let ABC be a triangle and angle B = 60. Angle bisectors AD and CE meet at the point O. Prove that OD = OE.
(I hope you can draw a diagram for this.)

Let $\angle BAD = \angle DAC = x^o$ and $\angle ACE = \angle ECB = y^o$

Then, $2x+2y+ 60=180$ (angle sum of $\triangle ABC$)

$\Rightarrow x+y=60$

Now $\angle DOC$ is an exterior angle of $\triangle OAC$ = sum of interior opposite angles $= x+y = 60^o$

$\Rightarrow BEOD$ is a cyclic quadrilateral (exterior angle $\angle DOC$ = interior opposite angle $\angle B$)

Now the angle bisectors of a triangle are concurrent $\Rightarrow BO$ is the bisector of $\angle ABC$

$\Rightarrow \angle DBO = \angle EBO$

$\Rightarrow \angle DEO = \angle EDO$ (angles in same segment)

$\Rightarrow OE = OD$ (isosceles $\triangle OED$)
user
The text contains two geometry problems, each with a detailed solution provided. The first problem involves proving the equality of two angles, and the second problem involves proving the equality of two segments in a triangle. Diagrams are requested and provided for the first problem, and the solution for the second problem is also given.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS