MHB Proving Perpendicularity of Two Segments

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The discussion focuses on proving the perpendicularity of two segments, specifically showing that angles XYA and BXE are equal. It is established that triangle XYA is isosceles due to congruent sides AY and AX, leading to the conclusion that angles XYA and AXY are equal. The relationship between angles BXE and BAD is also explored, with the use of triangle congruence to demonstrate that angle ADC is 90 degrees. Ultimately, it is confirmed that segments XE and AD are perpendicular to BC, reinforcing the proof of their relationship. The conversation emphasizes the importance of angle relationships and triangle properties in geometric proofs.
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Problem

View attachment 6018

Workings

I was able to show that the triangles ABD & ACD are congruent as $AD$ is a common side and as $\angle BAC $ is bisected & $\angle ABC = \angle ACB$ give which makes them congruent by Angle Angle Side.

Where do I need help:

I am having trouble showing,

Show that $\angle XYA = \angle BXE$

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& $\angle BEX = \angle BXE + \angle EBX$
 

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Hi mathlearn,

Considering your first question, it's given that $AY$ and $AX$ are congruent, making triangle $XYA$ isosceles. What then can you say about $\angle XYA$ and $\angle AXY$? Also, what's the relationship between angles $\angle AXY$ and $\angle BXE$?

As for your second question, consider triangle $CYE$. The measure of exterior angle $\angle YEB$ is equal to the sum of the measures of remote interior angles $\angle CYE$ and $\angle ECY$: $\measuredangle YEB = \measuredangle CYE + \measuredangle ECY$. It's given that $\angle ABC\cong \angle ACB$, and by a previous part $\measuredangle XYA = \measuredangle BXE$. By substitution you'll get $\measuredangle BEX = \measuredangle BXE + \measuredangle EBX$.
 
Euge said:
Hi mathlearn,

Considering your first question, it's given that $AY$ and $AX$ are congruent, making triangle $XYA$ isosceles. What then can you say about $\angle XYA$ and $\angle AXY$? Also, what's the relationship between angles $\angle AXY$ and $\angle BXE$?

Bringing in the diagram again here,

View attachment 6021

$\angle XYA$ = $\angle AXY$ = $\angle BXE \left(\text{vertically opposite angles}\right)$

Euge said:
Hi mathlearn,

As for your second question, consider triangle $CYE$. The measure of exterior angle $\angle YEB$ is equal to the sum of the measures of remote interior angles $\angle CYE$ and $\angle ECY$: $\measuredangle YEB = \measuredangle CYE + \measuredangle ECY$. It's given that $\angle ABC\cong \angle ACB$, and by a previous part $\measuredangle XYA = \measuredangle BXE$. By substitution you'll get $\measuredangle BEX = \measuredangle BXE + \measuredangle EBX$.

Thank you very much (Nod)

For the third question as $\measuredangle ADC$ which is 90 = $\measuredangle XEB$ perpendicular drawn , therefore XE//AD
 

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mathlearn said:
Bringing in the diagram again here,
$\angle XYA$ = $\angle AXY$ = $\angle BXE \left(\text{vertically opposite angles}\right)$
Thank you very much (Nod)

For the third question as $\measuredangle ADC$ which is 90 = $\measuredangle XEB$ perpendicular drawn , therefore XE//AD

I was just taking a look again at this problem and how can we say that $\measuredangle XEB$ = 90 degrees , I know that $\measuredangle XEB$ = 90 degrees but how can we prove that ? as nowhere in the problem mentions that line XE is a perpendicular drawn

$\measuredangle ADC$ can be proved 90 using triangle congruence

Many Thanks :)
 
Hi mathlearn,

It does turn out in this case that $\overline{XE}$ and $\overline{AD}$ are perpendicular to $\overline{BC}$, but by corresponding angles, to show that $\overline{XE}\,\|\, \overline{AD}$, it is enough to show that $m\measuredangle BXE = m\measuredangle BAD$. Let $\alpha = m\measuredangle XYA$ and $\beta= m\measuredangle XBE$. By the remote interior angles theorem $m\measuredangle XAY = m\measuredangle XBE + m\measuredangle ACB = \beta + \beta = 2\beta$. Since the sum of the measures of the interior angles of triangle $XAY$ (the measures are $\alpha, \alpha$, and $2\beta$) is $180^\circ$, then $2\alpha + 2\beta = 180^\circ$. Thus, looking to triangle $ABC$, we find $m\measuredangle BAC = 2\beta$. Since $\overline{AD}$ bisects $\measuredangle BAC$, $m\measuredangle BAD = \alpha$. We also know that $m\measuredangle BXE = \alpha$ from a previous exercise, so $m\measuredangle BAD = m\measuredangle BXE$, as desired.

Now, if you want to prove perpendicularity of $\overline{XE}$ and $\overline{AD}$ (and hence that $\overline{XE}\, \| \,\overline{AD}$), use the fact that $\alpha + \beta = 90^\circ$ (which follows from the equation $2\alpha + 2\beta = 180^\circ$ above). Since $\alpha = m\measuredangle BXE$, looking to triangle $BXE$ we find $m\measuredangle XEB = 90^\circ$; as $\alpha = m\measuredangle BAD$ (shown above), looking to triangle $BAD$ we find $m\measuredangle ADB = 90^\circ$. Hence, $\overline{XE}$ and $\overline{AD}$ are perpendicular to $\overline{BC}$.
 
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