In the diagram ,N lies on a side of the square ABCD,AM and LC are perpendicular to DN,prove that ADN=LCD.
Given :
N lies on a side of the square ABCD,AM and LC are perpendicular to DN,
To Find : prove that ∠ADN= ∠LCD.
Solution
Let say
∠ADN= α
∠LCD = β
∠ADN + ∠CDN = 90° ( ∠D = angle of square )
=> α + ∠CDN = 90°
∠CDN = ∠CDL
=> α + ∠CDL = 90°
in ΔCDL
∠CDL + ∠LCD + ∠DLC = 180° ( sum of angles of triangle )
∠DLC = 90° as LC is perpendicular to DN
=> ∠CDL + β + 90° = 180°
=> ∠CDL + β = 90°
α + ∠CDL = 90°
∠CDL + β = 90°
Equating both
α + ∠CDL = ∠CDL + β
=> α = β
=> ∠ADN= ∠LCD
QED
Hence Proved
Well, well, well! Looks like we have ourselves a geometry question. Let's see if we can untangle this one with a dash of humor!
Now, we're given that N lies on a side of the square ABCD, and that AM and LC are perpendicular to DN. We need to prove that ADN is equal to LCD.
Let's get this square dance started, shall we?
First, since ABCD is a square, we know that all its sides are equal. Hmm, ABCD sounds like a catchy tune - "All Boys Can Dance!"
Now, since AM and LC are perpendicular to DN, it means that AM is parallel to LN. Sounds like these lines are really getting their groove on!
So, we have two pairs of parallel lines - AMLN and ADNC. That reminds me of a circus act - "The Amazing Parallel Lines!"
And hey, guess what? Since we have parallel lines, we also have some equal angles! That's right, the angles formed by AM and LN are equal, as well as the angles formed by AD and NC.
Now, let's bring it all together! Since ABCD is a square, we know that all its angles are equal. So, the angles ADN and NDC are equal, just like the angles LCD and LAN.
And ta-da! We have proved that ADN is equal to LCD. It's like they're doing a synchronized dance routine – the "Angle Equality Shuffle!"
So, my friend, that's how you prove ADN equals LCD. With a sprinkle of humor and a whole lot of parallel lines, we've got ourselves a fantastic geometry marvel! Keep up the good work!
To prove that angle ADN is congruent to angle LCD, we can use the fact that the opposite sides of a square are parallel and equal in length.
Step 1: Given that N lies on a side of the square ABCD, we can assume that DN is one of the sides of the square.
Step 2: Since AM and LC are perpendicular to DN, we can conclude that angle MAD is congruent to angle LDC. This is because angles formed by a line intersecting two parallel lines, in this case, AM and LC, are congruent.
Step 3: We know that a square has four 90-degree angles, so angle ADN is 90 degrees.
Step 4: Similarly, a square also has four 90-degree angles, so angle LCD is also 90 degrees.
Step 5: Since angles ADN and LCD are both 90 degrees, they are congruent.
Therefore, we have proven that angle ADN is congruent to angle LCD.
To prove that ∠ADN = ∠LCD, we need to show that the triangle ADN and the triangle LCD are similar.
Here's how you can prove it:
Step 1: Take a look at the given diagram and identify the important information. In this case, we are given that N lies on a side of the square ABCD, and AM and LC are perpendicular to DN.
Step 2: From the given information, we can observe that AM and LC are altitudes of the triangles ADN and LCD, respectively.
Step 3: Since AM and LC are altitudes, we know that they are perpendicular to the sides DN and NC, respectively. Therefore, DN is perpendicular to AM, and NC is perpendicular to LC.
Step 4: Now, let's focus on the triangles ADN and LCD. We already know that DN is perpendicular to AM and NC is perpendicular to LC. This means that the triangles ADN and LCD share a common angle, which is the right angle at M.
Step 5: By definition, if two triangles have one angle in common, and the sides including this angle are perpendicular, then the triangles are similar.
Step 6: Therefore, we can conclude that the triangles ADN and LCD are similar.
Step 7: Since the triangles ADN and LCD are similar, the corresponding angles will be equal.
Step 8: Thus, we have proven that ∠ADN = ∠LCD.
Note: It's important to understand the properties of right triangles, altitudes, and similar triangles to successfully prove this statement.