Prove the given geometry problems. Problem 1: In ABC, AE=EB and EF||BC. In ACD, FG||CD. Prove AG=GD. Problem 2: In DEF, DS=SE, EU=UF, and ST||EF. Prove STUE is a parallelogram.

Hey 🥹❤️‍🔥M💞🥹Miseketeri, good to see you again. Here are the solutions to your geometry problems. Problem 1: Given: 1. In ABC, AE = EB and EF BC. 2. In ACD, FG CD. Prove: AG = GD. Step 1: Analyze ABC. We are given that AE = EB, which means E is the midpoint of side AB. We are also given that EF BC. By the converse of the Midpoint Theorem (also known as Thales's Theorem or Basic Proportionality Theorem), if a line parallel to one side of a triangle passes through the midpoint of another side, then it bisects the third side. Therefore, F must be the midpoint of side AC. This implies AF = FC. Step 2: Analyze ACD. From Step 1, we established that F is the midpoint of side AC. We are given that FG CD. Applying the converse of the Midpoint Theorem to ACD: Since FG CD and F is the midpoint of AC, then G must be the midpoint of side AD. Therefore, AG = GD. This completes the proof. AG = GD Problem 2: Given: In DEF, DS = SE, EU = UF, and ST EF. Prove: STUE is a parallelogram. Step 1: Analyze DEF using the given information about S. We are given that DS = SE, which means S is the midpoint of side DE. We are also given that ST EF. By the converse of the Midpoint Theorem, if a line parallel to one side of a triangle passes through the midpoint of another side, then it bisects the third side. Therefore, T must be the midpoint of side DF. Step 2: Apply the Midpoint Theorem to DEF. Since S is the midpoint of DE and T is the midpoint of DF, by the Midpoint Theorem, the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half the length of the third side. So, ST = (1)/(2) EF. Step 3: Use the given information about U. We are given that EU = UF, which means U is the midpoint of side EF. Therefore, EU = (1)/(2) EF. Step 4: Compare ST and EU. From Step 2, ST = (1)/(2) EF. From Step 3, EU = (1)/(2) EF. Thus, ST = EU. Step 5: Establish parallelism. We are given that ST EF. Since U lies on EF, it follows that ST EU. Step 6: Conclude that STUE is a parallelogram. We have shown that in quadrilateral STUE, one pair of opposite sides (ST and EU) are both parallel (ST EU) and equal in length (ST = EU). A quadrilateral with one pair of opposite sides that are both parallel and equal in length is a parallelogram. Therefore, STUE is a parallelogram. This completes the proof. STUE is a parallelogram Send me the next one 📸